Sr Examen
Otras calculadoras
- Derivada de una función implícitamente dada
- Derivada de una función paramétrica
- Derivada parcial de la función
- Análisis de la función gráfica
- Integrales paso a paso
- Ecuaciones diferenciales paso a paso
- Límites paso por paso
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¿Cómo usar?
- Derivada de:
-
Derivada de -2*y
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Derivada de 3^2*x
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Derivada de 2*x+8/x
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Derivada de -1/y
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Expresiones idénticas
- y=x*arcsin(uno /x)+ln(abs(x+sqrt(x^ dos - uno)))
- y es igual a x multiplicar por arc seno de (1 dividir por x) más ln(abs(x más raíz cuadrada de (x al cuadrado menos 1)))
- y es igual a x multiplicar por arc seno de (uno dividir por x) más ln(abs(x más raíz cuadrada de (x en el grado dos menos uno)))
- y=x*arcsin(1/x)+ln(abs(x+√(x^2-1)))
- y=x*arcsin(1/x)+ln(abs(x+sqrt(x2-1)))
- y=x*arcsin1/x+lnabsx+sqrtx2-1
- y=x*arcsin(1/x)+ln(abs(x+sqrt(x²-1)))
- y=x*arcsin(1/x)+ln(abs(x+sqrt(x en el grado 2-1)))
- y=xarcsin(1/x)+ln(abs(x+sqrt(x^2-1)))
- y=xarcsin(1/x)+ln(abs(x+sqrt(x2-1)))
- y=xarcsin1/x+lnabsx+sqrtx2-1
- y=xarcsin1/x+lnabsx+sqrtx^2-1
- y=x*arcsin(1 dividir por x)+ln(abs(x+sqrt(x^2-1)))
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Expresiones semejantes
- y=x*arcsin(1/x)-ln(abs(x+sqrt(x^2-1)))
- y=x*arcsin(1/x)+ln(abs(x+sqrt(x^2+1)))
- y=x*arcsin(1/x)+ln(abs(x-sqrt(x^2-1)))
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Expresiones con funciones
- ln
- ln(x+10)^11
- ln((x^2)-1)
- ln3x^5
- ln^3(x^2+1)
- ln(1+|x|)
- Raíz cuadrada sqrt
- sqrt(x*2)
- sqrt((x-1)/2)
- sqrt(3*y)
- sqrt(10*x)
- sqrt((26/25)^3)+(log(1.02))
Derivada de y=x*arcsin(1/x)+ln(abs(x+sqrt(x^2-1)))
Solución
Ha introducido
[src]
/| ________|\
/1\ || / 2 ||
x*asin|-| + log\|x + \/ x - 1 |/
\x/
$$x \operatorname{asin}{\left(\frac{1}{x} \right)} + \log{\left(\left|{x + \sqrt{x^{2} - 1}}\right| \right)}$$
x*asin(1/x) + log(Abs(x + sqrt(x^2 - 1)))
Primera derivada
[src]
// / / 2\\ \ \
|| |atan2\0, -1 + x /| / 2\| / _______________ \ |
|| x*cos|-----------------|*sign\-1 + x /| | / 2 / / 2\\| / / 2\\ | / _________\
|| \ 2 / | | / -1 + x |atan2\0, -1 + x /|| 2|atan2\0, -1 + x /| / 2\| | / 2 |
||1 + --------------------------------------|*|x + / ------------- *cos|-----------------|| + x*sin |-----------------|*sign\-1 + x /|*sign\x + \/ -1 + x /
|| _______________ | | / / 2\ \ 2 /| \ 2 / |
|| / 2 | \ \/ sign\-1 + x / / |
|| / -1 + x | |
|| / ------------- | |
|| / / 2\ | |
1 \\ \/ sign\-1 + x / / / /1\
- --------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------------ + asin|-|
________ / _________\ | ________| \x/
/ 1 | / 2 | | / 2 |
x* / 1 - -- \x + \/ -1 + x /*|x + \/ x - 1 |
/ 2
\/ x
$$\operatorname{asin}{\left(\frac{1}{x} \right)} + \frac{\left(x \sin^{2}{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)} \operatorname{sign}{\left(x^{2} - 1 \right)} + \left(x + \sqrt{\frac{x^{2} - 1}{\operatorname{sign}{\left(x^{2} - 1 \right)}}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)}\right) \left(\frac{x \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)} \operatorname{sign}{\left(x^{2} - 1 \right)}}{\sqrt{\frac{x^{2} - 1}{\operatorname{sign}{\left(x^{2} - 1 \right)}}}} + 1\right)\right) \operatorname{sign}{\left(x + \sqrt{x^{2} - 1} \right)}}{\left(x + \sqrt{x^{2} - 1}\right) \left|{x + \sqrt{x^{2} - 1}}\right|} - \frac{1}{x \sqrt{1 - \frac{1}{x^{2}}}}$$
Segunda derivada
[src]
/ / / / 2\ / 2\\ \ \
| | 2 | 2*\-1 + x /*DiracDelta\-1 + x /| / 2\ | |
| / _______________ \ / _______________ \ | x *|-1 + -------------------------------|*sign\-1 + x / | |
| | / 2 / / 2\ / 2\\ / / 2\\| | / 2 / / 2\\| | | / 2\ | | / / 2\\| 2
// / / 2\\ \ \ | / / / 2\\ \ | / -1 + x | 2*\-1 + x /*DiracDelta\-1 + x /| |atan2\0, -1 + x /|| | / -1 + x |atan2\0, -1 + x /|| | 2 / 2\ \ sign\-1 + x / / / 2\| |atan2\0, -1 + x /|| // / / 2\\ \ \ // / / 2\\ \ \
|| |atan2\0, -1 + x /| / 2\| / _______________ \ | | | |atan2\0, -1 + x /| / 2\| | x* / ------------- *|-1 + -------------------------------|*cos|-----------------|| |x + / ------------- *cos|-----------------||*|4*x *DiracDelta\-1 + x / + ------------------------------------------------------- + sign\-1 + x /|*cos|-----------------|| || |atan2\0, -1 + x /| / 2\| / _______________ \ | || |atan2\0, -1 + x /| / 2\| / _______________ \ |
|| x*cos|-----------------|*sign\-1 + x /| | / 2 / / 2\\| / / 2\\ | / / _________\\ | / / 2\\ | x*cos|-----------------|*sign\-1 + x /| | / / 2\ | / 2\ | \ 2 /| / / 2\\ | / / 2\ \ 2 /| | 2 | \ 2 /| / _________\ || x*cos|-----------------|*sign\-1 + x /| | / 2 / / 2\\| / / 2\\ | / _________\ || x*cos|-----------------|*sign\-1 + x /| | / 2 / / 2\\| / / 2\\ | / _________\
|| \ 2 / | | / -1 + x |atan2\0, -1 + x /|| 2|atan2\0, -1 + x /| / 2\| d | | / 2 || | 2|atan2\0, -1 + x /| / 2\ | \ 2 / | | \/ sign\-1 + x / \ sign\-1 + x / / | 2 2|atan2\0, -1 + x /| / 2\ \ \/ sign\-1 + x / / \ -1 + x / | | / 2 | || \ 2 / | | / -1 + x |atan2\0, -1 + x /|| 2|atan2\0, -1 + x /| / 2\| 2| / 2 | / x \ || \ 2 / | | / -1 + x |atan2\0, -1 + x /|| 2|atan2\0, -1 + x /| / 2\| | / 2 |
||1 + --------------------------------------|*|x + / ------------- *cos|-----------------|| + x*sin |-----------------|*sign\-1 + x /|*--\sign\x + \/ -1 + x // |sin |-----------------|*sign\-1 + x / - |1 + --------------------------------------|*|-1 + -------------------------------------------------------------------------------------| + 4*x *sin |-----------------|*DiracDelta\-1 + x / + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|*sign\x + \/ -1 + x / ||1 + --------------------------------------|*|x + / ------------- *cos|-----------------|| + x*sin |-----------------|*sign\-1 + x /| *sign \x + \/ -1 + x / |1 + ------------|*||1 + --------------------------------------|*|x + / ------------- *cos|-----------------|| + x*sin |-----------------|*sign\-1 + x /|*sign\x + \/ -1 + x /
|| _______________ | | / / 2\ \ 2 /| \ 2 / | dx | \ 2 / | _______________ | | 2 | \ 2 / _______________ | || _______________ | | / / 2\ \ 2 /| \ 2 / | | _________| || _______________ | | / / 2\ \ 2 /| \ 2 / |
|| / 2 | \ \/ sign\-1 + x / / | | | / 2 | \ -1 + x / / 2 | || / 2 | \ \/ sign\-1 + x / / | | / 2 | || / 2 | \ \/ sign\-1 + x / / |
|| / -1 + x | | | | / -1 + x | / -1 + x | || / -1 + x | | \ \/ -1 + x / || / -1 + x | |
|| / ------------- | | | | / ------------- | / ------------- | || / ------------- | | || / ------------- | |
|| / / 2\ | | | | / / 2\ | / / 2\ | || / / 2\ | | || / / 2\ | |
1 \\ \/ sign\-1 + x / / / \ \ \/ sign\-1 + x / / \/ sign\-1 + x / / \\ \/ sign\-1 + x / / / \\ \/ sign\-1 + x / / /
-------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
3/2 / _________\ | _________| / _________\ | _________| 2 2 2
4 / 1 \ | / 2 | | / 2 | | / 2 | | / 2 | / _________\ | _________| / _________\ | _________|
x *|1 - --| \x + \/ -1 + x /*|x + \/ -1 + x | \x + \/ -1 + x /*|x + \/ -1 + x | | / 2 | | / 2 | | / 2 | | / 2 |
| 2| \x + \/ -1 + x / *|x + \/ -1 + x | \x + \/ -1 + x / *|x + \/ -1 + x |
\ x /
$$\frac{\left(x \sin^{2}{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)} \operatorname{sign}{\left(x^{2} - 1 \right)} + \left(x + \sqrt{\frac{x^{2} - 1}{\operatorname{sign}{\left(x^{2} - 1 \right)}}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)}\right) \left(\frac{x \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)} \operatorname{sign}{\left(x^{2} - 1 \right)}}{\sqrt{\frac{x^{2} - 1}{\operatorname{sign}{\left(x^{2} - 1 \right)}}}} + 1\right)\right) \frac{d}{d x} \operatorname{sign}{\left(x + \sqrt{x^{2} - 1} \right)}}{\left(x + \sqrt{x^{2} - 1}\right) \left|{x + \sqrt{x^{2} - 1}}\right|} + \frac{\left(4 x^{2} \sin^{2}{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)} \delta\left(x^{2} - 1\right) - \left(\frac{x \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)} \operatorname{sign}{\left(x^{2} - 1 \right)}}{\sqrt{\frac{x^{2} - 1}{\operatorname{sign}{\left(x^{2} - 1 \right)}}}} + 1\right) \left(\frac{x \sqrt{\frac{x^{2} - 1}{\operatorname{sign}{\left(x^{2} - 1 \right)}}} \left(\frac{2 \left(x^{2} - 1\right) \delta\left(x^{2} - 1\right)}{\operatorname{sign}{\left(x^{2} - 1 \right)}} - 1\right) \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)}}{x^{2} - 1} - 1\right) + \sin^{2}{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)} \operatorname{sign}{\left(x^{2} - 1 \right)} + \frac{\left(x + \sqrt{\frac{x^{2} - 1}{\operatorname{sign}{\left(x^{2} - 1 \right)}}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)}\right) \left(4 x^{2} \delta\left(x^{2} - 1\right) + \frac{x^{2} \left(\frac{2 \left(x^{2} - 1\right) \delta\left(x^{2} - 1\right)}{\operatorname{sign}{\left(x^{2} - 1 \right)}} - 1\right) \operatorname{sign}{\left(x^{2} - 1 \right)}}{x^{2} - 1} + \operatorname{sign}{\left(x^{2} - 1 \right)}\right) \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)}}{\sqrt{\frac{x^{2} - 1}{\operatorname{sign}{\left(x^{2} - 1 \right)}}}}\right) \operatorname{sign}{\left(x + \sqrt{x^{2} - 1} \right)}}{\left(x + \sqrt{x^{2} - 1}\right) \left|{x + \sqrt{x^{2} - 1}}\right|} - \frac{\left(\frac{x}{\sqrt{x^{2} - 1}} + 1\right) \left(x \sin^{2}{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)} \operatorname{sign}{\left(x^{2} - 1 \right)} + \left(x + \sqrt{\frac{x^{2} - 1}{\operatorname{sign}{\left(x^{2} - 1 \right)}}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)}\right) \left(\frac{x \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)} \operatorname{sign}{\left(x^{2} - 1 \right)}}{\sqrt{\frac{x^{2} - 1}{\operatorname{sign}{\left(x^{2} - 1 \right)}}}} + 1\right)\right) \operatorname{sign}{\left(x + \sqrt{x^{2} - 1} \right)}}{\left(x + \sqrt{x^{2} - 1}\right)^{2} \left|{x + \sqrt{x^{2} - 1}}\right|} - \frac{\left(x \sin^{2}{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)} \operatorname{sign}{\left(x^{2} - 1 \right)} + \left(x + \sqrt{\frac{x^{2} - 1}{\operatorname{sign}{\left(x^{2} - 1 \right)}}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)}\right) \left(\frac{x \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)} \operatorname{sign}{\left(x^{2} - 1 \right)}}{\sqrt{\frac{x^{2} - 1}{\operatorname{sign}{\left(x^{2} - 1 \right)}}}} + 1\right)\right)^{2} \operatorname{sign}^{2}{\left(x + \sqrt{x^{2} - 1} \right)}}{\left(x + \sqrt{x^{2} - 1}\right)^{2} \left|{x + \sqrt{x^{2} - 1}}\right|^{2}} + \frac{1}{x^{4} \left(1 - \frac{1}{x^{2}}\right)^{\frac{3}{2}}}$$
Tercera derivada
[src]
/ / 2 \ \
| | / / 2\ / 2\\ / / 2\ / 2\\ / / 2\ / 2\\ | |
| / / / 2\\ \ | 2 | 2*\-1 + x /*DiracDelta\-1 + x /| 2 | 2*\-1 + x /*DiracDelta\-1 + x /| 2 | 2*\-1 + x /*DiracDelta\-1 + x /| / 2\| |
| / 2 \ _______________ | |atan2\0, -1 + x /| / 2\| | x *|-1 + -------------------------------| 2*x *|-1 + -------------------------------| 4*x *|-1 + -------------------------------|*DiracDelta\-1 + x /| |
| / _______________ \ | / / 2\ / 2\ 2 / 2\ 2 2/ 2\ / 2\ 2 / 2\ / 2 \\ / / 2\ / 2\\ / / 2\ / 2\\ / / 2\ / 2\\ / / 2\ / 2\\ | / 2 | x*cos|-----------------|*sign\-1 + x /| | | / 2\ | | / 2\ | / 2\ / 2\ 2 / 2\ 2 2/ 2\ / 2\ 2 / 2\ / 2 \ | / 2\ | | / / 2\\|
| | / 2 / / 2\ / 2\\ / / 2\\| / / / 2\ / 2\\ \ | | 2*\-1 + x /*DiracDelta\-1 + x / 8*x *DiracDelta\-1 + x / 16*x *DiracDelta \-1 + x /*\-1 + x / 4*x *\-1 + x /*DiracDelta\-1 + x , 1/| / 2\ | 2*\-1 + x /*DiracDelta\-1 + x /| / 2\ 2 | 2*\-1 + x /*DiracDelta\-1 + x /| / 2\ 2 | 2*\-1 + x /*DiracDelta\-1 + x /| / 2\ 2 | 2*\-1 + x /*DiracDelta\-1 + x /| / 2\| / -1 + x | \ 2 / | | \ sign\-1 + x / / \ sign\-1 + x / / 2*\-1 + x /*DiracDelta\-1 + x / 8*x *DiracDelta\-1 + x / 16*x *DiracDelta \-1 + x /*\-1 + x / 4*x *\-1 + x /*DiracDelta\-1 + x , 1/ \ sign\-1 + x / / | |atan2\0, -1 + x /|| / / / / 2\ / 2\\ \ \ / / / / 2\ / 2\\ \ \ / / / / 2\ / 2\\ \ \
| | / -1 + x | 2*\-1 + x /*DiracDelta\-1 + x /| |atan2\0, -1 + x /|| | 2 | 2*\-1 + x /*DiracDelta\-1 + x /| / 2\ | / _______________ \ | |-1 + ------------------------------- + ------------------------ - ------------------------------------ + -------------------------------------|*sign\-1 + x / 2*|-1 + -------------------------------|*sign\-1 + x / x *|-1 + -------------------------------| *sign\-1 + x / 2*x *|-1 + -------------------------------|*sign\-1 + x / 12*x *|-1 + -------------------------------|*DiracDelta\-1 + x /| / ------------- *|1 + --------------------------------------|*|-1 - ------------------------------------------ - ------------------------------------------- + ------------------------------- + ------------------------ - ------------------------------------ + ------------------------------------- + ---------------------------------------------------------------|*cos|-----------------|| | | 2 | 2*\-1 + x /*DiracDelta\-1 + x /| / 2\ | | | | 2 | 2*\-1 + x /*DiracDelta\-1 + x /| / 2\ | | | | 2 | 2*\-1 + x /*DiracDelta\-1 + x /| / 2\ | |
| | x* / ------------- *|-1 + -------------------------------|*cos|-----------------|| | x *|-1 + -------------------------------|*sign\-1 + x / | | / 2 / / 2\\| | | / 2\ / 2\ 2/ 2\ / 2\ | | / 2\ | | / 2\ | | / 2\ | | / 2\ | | / / 2\\ / / 2\ | _______________ | | 2 2 / 2\ / 2\ 2/ 2\ / 2\ / 2\ | \ 2 /| | / _______________ \ / _______________ \ | x *|-1 + -------------------------------|*sign\-1 + x / | | | / _______________ \ / _______________ \ | x *|-1 + -------------------------------|*sign\-1 + x / | | | / _______________ \ / _______________ \ | x *|-1 + -------------------------------|*sign\-1 + x / | |
| | / / 2\ | / 2\ | \ 2 /| | | / 2\ | | / / 2\\ | / -1 + x |atan2\0, -1 + x /|| | / 2\ 2 / 2 \ \ sign\-1 + x / sign\-1 + x / sign \-1 + x / sign\-1 + x / / \ sign\-1 + x / / \ sign\-1 + x / / \ sign\-1 + x / / \ sign\-1 + x / / | |atan2\0, -1 + x /| \/ sign\-1 + x / | / 2 | \ -1 + x -1 + x sign\-1 + x / sign\-1 + x / sign \-1 + x / sign\-1 + x / sign\-1 + x / / | | | / 2 / / 2\ / 2\\ / / 2\\| | / 2 / / 2\\| | | / 2\ | | / / 2\\| 3 2 | | / 2 / / 2\ / 2\\ / / 2\\| | / 2 / / 2\\| | | / 2\ | | / / 2\\| | | / 2 / / 2\ / 2\\ / / 2\\| | / 2 / / 2\\| | | / 2\ | | / / 2\\| 2
// / / 2\\ \ \ | | \/ sign\-1 + x / \ sign\-1 + x / / | | 2 / 2\ \ sign\-1 + x / / / 2\| |atan2\0, -1 + x /| x*|x + / ------------- *cos|-----------------||*|12*DiracDelta\-1 + x / + 8*x *DiracDelta\-1 + x , 1/ + -------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------ + -------------------------------------------------------- - --------------------------------------------------------- + ----------------------------------------------------------------|*cos|-----------------| | / -1 + x | | | / / / 2\\ \ | / -1 + x | 2*\-1 + x /*DiracDelta\-1 + x /| |atan2\0, -1 + x /|| | / -1 + x |atan2\0, -1 + x /|| | 2 / 2\ \ sign\-1 + x / / / 2\| |atan2\0, -1 + x /|| // / / 2\\ \ \ // / / 2\\ \ \ // / / 2\\ \ \ | / / / 2\\ \ | / -1 + x | 2*\-1 + x /*DiracDelta\-1 + x /| |atan2\0, -1 + x /|| | / -1 + x |atan2\0, -1 + x /|| | 2 / 2\ \ sign\-1 + x / / / 2\| |atan2\0, -1 + x /|| // / / 2\\ \ \ | / / / 2\\ \ | / -1 + x | 2*\-1 + x /*DiracDelta\-1 + x /| |atan2\0, -1 + x /|| | / -1 + x |atan2\0, -1 + x /|| | 2 / 2\ \ sign\-1 + x / / / 2\| |atan2\0, -1 + x /|| // / / 2\\ \ \ // / / 2\\ \ \ // / / 2\\ \ \
|| |atan2\0, -1 + x /| / 2\| / _______________ \ | | 2*|-1 + -------------------------------------------------------------------------------------|*|4*x *DiracDelta\-1 + x / + ------------------------------------------------------- + sign\-1 + x /|*cos|-----------------| | / / 2\ \ 2 /| | 2 2 2 2 2 | \ 2 / | / ------------- | | | | |atan2\0, -1 + x /| / 2\| | x* / ------------- *|-1 + -------------------------------|*cos|-----------------|| |x + / ------------- *cos|-----------------||*|4*x *DiracDelta\-1 + x / + ------------------------------------------------------- + sign\-1 + x /|*cos|-----------------|| || |atan2\0, -1 + x /| / 2\| / _______________ \ | || |atan2\0, -1 + x /| / 2\| / _______________ \ | || |atan2\0, -1 + x /| / 2\| / _______________ \ | | | |atan2\0, -1 + x /| / 2\| | x* / ------------- *|-1 + -------------------------------|*cos|-----------------|| |x + / ------------- *cos|-----------------||*|4*x *DiracDelta\-1 + x / + ------------------------------------------------------- + sign\-1 + x /|*cos|-----------------|| || |atan2\0, -1 + x /| / 2\| / _______________ \ | | | |atan2\0, -1 + x /| / 2\| | x* / ------------- *|-1 + -------------------------------|*cos|-----------------|| |x + / ------------- *cos|-----------------||*|4*x *DiracDelta\-1 + x / + ------------------------------------------------------- + sign\-1 + x /|*cos|-----------------|| || |atan2\0, -1 + x /| / 2\| / _______________ \ | || |atan2\0, -1 + x /| / 2\| / _______________ \ | || |atan2\0, -1 + x /| / 2\| / _______________ \ |
|| x*cos|-----------------|*sign\-1 + x /| | / 2 / / 2\\| / / 2\\ | 2/ / _________\\ | / / 2\\ / / 2\\ | 2 | | 2 | \ 2 / \ \/ sign\-1 + x / / | -1 + x -1 + x / 2\ / 2\ -1 + x | | / / 2\ | | / _________\ | / / 2\\ | x*cos|-----------------|*sign\-1 + x /| | / / 2\ | / 2\ | \ 2 /| / / 2\\ | / / 2\ \ 2 /| | 2 | \ 2 /| / / _________\\ || x*cos|-----------------|*sign\-1 + x /| | / 2 / / 2\\| / / 2\\ | / _________\ || x*cos|-----------------|*sign\-1 + x /| | / 2 / / 2\\| / / 2\\ | / / _________\\ / _________\ / _________\ || x*cos|-----------------|*sign\-1 + x /| | / 2 / / 2\\| / / 2\\ | | / / 2\\ | x*cos|-----------------|*sign\-1 + x /| | / / 2\ | / 2\ | \ 2 /| / / 2\\ | / / 2\ \ 2 /| | 2 | \ 2 /| || x*cos|-----------------|*sign\-1 + x /| | / 2 / / 2\\| / / 2\\ | / / _________\\ | / / 2\\ | x*cos|-----------------|*sign\-1 + x /| | / / 2\ | / 2\ | \ 2 /| / / 2\\ | / / 2\ \ 2 /| | 2 | \ 2 /| / _________\ 2 || x*cos|-----------------|*sign\-1 + x /| | / 2 / / 2\\| / / 2\\ | / _________\ || x*cos|-----------------|*sign\-1 + x /| | / 2 / / 2\\| / / 2\\ | / _________\ / 2 \ || x*cos|-----------------|*sign\-1 + x /| | / 2 / / 2\\| / / 2\\ | / _________\
|| \ 2 / | | / -1 + x |atan2\0, -1 + x /|| 2|atan2\0, -1 + x /| / 2\| d | | / 2 || | 3 2|atan2\0, -1 + x /| / 2 \ 2|atan2\0, -1 + x /| / 2\ \ -1 + x / \ -1 + x / \ \-1 + x / \-1 + x / / \ \/ sign\-1 + x / / | | / 2 | | 2|atan2\0, -1 + x /| / 2\ | \ 2 / | | \/ sign\-1 + x / \ sign\-1 + x / / | 2 2|atan2\0, -1 + x /| / 2\ \ \/ sign\-1 + x / / \ -1 + x / | d | | / 2 || || \ 2 / | | / -1 + x |atan2\0, -1 + x /|| 2|atan2\0, -1 + x /| / 2\| 3| / 2 | || \ 2 / | | / -1 + x |atan2\0, -1 + x /|| 2|atan2\0, -1 + x /| / 2\| d | | / 2 || | / 2 | 2| / 2 | || \ 2 / | | / -1 + x |atan2\0, -1 + x /|| 2|atan2\0, -1 + x /| / 2\| | 2|atan2\0, -1 + x /| / 2\ | \ 2 / | | \/ sign\-1 + x / \ sign\-1 + x / / | 2 2|atan2\0, -1 + x /| / 2\ \ \/ sign\-1 + x / / \ -1 + x / | / x \ || \ 2 / | | / -1 + x |atan2\0, -1 + x /|| 2|atan2\0, -1 + x /| / 2\| d | | / 2 || / x \ | 2|atan2\0, -1 + x /| / 2\ | \ 2 / | | \/ sign\-1 + x / \ sign\-1 + x / / | 2 2|atan2\0, -1 + x /| / 2\ \ \/ sign\-1 + x / / \ -1 + x / | | / 2 | / x \ || \ 2 / | | / -1 + x |atan2\0, -1 + x /|| 2|atan2\0, -1 + x /| / 2\| | / 2 | || \ 2 / | | / -1 + x |atan2\0, -1 + x /|| 2|atan2\0, -1 + x /| / 2\| 2| / 2 | / x \ | x | || \ 2 / | | / -1 + x |atan2\0, -1 + x /|| 2|atan2\0, -1 + x /| / 2\| | / 2 |
||1 + --------------------------------------|*|x + / ------------- *cos|-----------------|| + x*sin |-----------------|*sign\-1 + x /|*---\sign\x + \/ -1 + x // |8*x *sin |-----------------|*DiracDelta\-1 + x , 1/ + 12*x*sin |-----------------|*DiracDelta\-1 + x / - -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ - -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|*sign\x + \/ -1 + x / 2*|sin |-----------------|*sign\-1 + x / - |1 + --------------------------------------|*|-1 + -------------------------------------------------------------------------------------| + 4*x *sin |-----------------|*DiracDelta\-1 + x / + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|*--\sign\x + \/ -1 + x // 2*||1 + --------------------------------------|*|x + / ------------- *cos|-----------------|| + x*sin |-----------------|*sign\-1 + x /| *sign \x + \/ -1 + x / 3*||1 + --------------------------------------|*|x + / ------------- *cos|-----------------|| + x*sin |-----------------|*sign\-1 + x /| *--\sign\x + \/ -1 + x //*sign\x + \/ -1 + x / 3*sign \x + \/ -1 + x /*||1 + --------------------------------------|*|x + / ------------- *cos|-----------------|| + x*sin |-----------------|*sign\-1 + x /|*|sin |-----------------|*sign\-1 + x / - |1 + --------------------------------------|*|-1 + -------------------------------------------------------------------------------------| + 4*x *sin |-----------------|*DiracDelta\-1 + x / + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| 2*|1 + ------------|*||1 + --------------------------------------|*|x + / ------------- *cos|-----------------|| + x*sin |-----------------|*sign\-1 + x /|*--\sign\x + \/ -1 + x // 2*|1 + ------------|*|sin |-----------------|*sign\-1 + x / - |1 + --------------------------------------|*|-1 + -------------------------------------------------------------------------------------| + 4*x *sin |-----------------|*DiracDelta\-1 + x / + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|*sign\x + \/ -1 + x / 2*|1 + ------------| *||1 + --------------------------------------|*|x + / ------------- *cos|-----------------|| + x*sin |-----------------|*sign\-1 + x /|*sign\x + \/ -1 + x / 3*||1 + --------------------------------------|*|x + / ------------- *cos|-----------------|| + x*sin |-----------------|*sign\-1 + x /| *sign \x + \/ -1 + x /*|1 + ------------| |-1 + -------|*||1 + --------------------------------------|*|x + / ------------- *cos|-----------------|| + x*sin |-----------------|*sign\-1 + x /|*sign\x + \/ -1 + x /
|| _______________ | | / / 2\ \ 2 /| \ 2 / | 2 | \ 2 / \ 2 / _______________ _______________ 2 | | \ 2 / | _______________ | | 2 | \ 2 / _______________ | dx || _______________ | | / / 2\ \ 2 /| \ 2 / | || _______________ | | / / 2\ \ 2 /| \ 2 / | dx || _______________ | | / / 2\ \ 2 /| \ 2 / | | \ 2 / | _______________ | | 2 | \ 2 / _______________ | | _________| || _______________ | | / / 2\ \ 2 /| \ 2 / | dx | _________| | \ 2 / | _______________ | | 2 | \ 2 / _______________ | | _________| || _______________ | | / / 2\ \ 2 /| \ 2 / | || _______________ | | / / 2\ \ 2 /| \ 2 / | | _________| | 2| || _______________ | | / / 2\ \ 2 /| \ 2 / |
|| / 2 | \ \/ sign\-1 + x / / | dx | / 2 / 2 -1 + x | | | / 2 | \ -1 + x / / 2 | || / 2 | \ \/ sign\-1 + x / / | || / 2 | \ \/ sign\-1 + x / / | || / 2 | \ \/ sign\-1 + x / / | | | / 2 | \ -1 + x / / 2 | | / 2 | || / 2 | \ \/ sign\-1 + x / / | | / 2 | | | / 2 | \ -1 + x / / 2 | | / 2 | || / 2 | \ \/ sign\-1 + x / / | || / 2 | \ \/ sign\-1 + x / / | | / 2 | \ -1 + x / || / 2 | \ \/ sign\-1 + x / / |
|| / -1 + x | | | / -1 + x / -1 + x | | | / -1 + x | / -1 + x | || / -1 + x | | || / -1 + x | | || / -1 + x | | | | / -1 + x | / -1 + x | \ \/ -1 + x / || / -1 + x | | \ \/ -1 + x / | | / -1 + x | / -1 + x | \ \/ -1 + x / || / -1 + x | | || / -1 + x | | \ \/ -1 + x / || / -1 + x | |
|| / ------------- | | | / ------------- / ------------- | | | / ------------- | / ------------- | || / ------------- | | || / ------------- | | || / ------------- | | | | / ------------- | / ------------- | || / ------------- | | | | / ------------- | / ------------- | || / ------------- | | || / ------------- | | || / ------------- | |
|| / / 2\ | | | / / 2\ / / 2\ | | | / / 2\ | / / 2\ | || / / 2\ | | || / / 2\ | | || / / 2\ | | | | / / 2\ | / / 2\ | || / / 2\ | | | | / / 2\ | / / 2\ | || / / 2\ | | || / / 2\ | | || / / 2\ | |
4 3 \\ \/ sign\-1 + x / / / \ \/ sign\-1 + x / \/ sign\-1 + x / / \ \ \/ sign\-1 + x / / \/ sign\-1 + x / / \\ \/ sign\-1 + x / / / \\ \/ sign\-1 + x / / / \\ \/ sign\-1 + x / / / \ \ \/ sign\-1 + x / / \/ sign\-1 + x / / \\ \/ sign\-1 + x / / / \ \ \/ sign\-1 + x / / \/ sign\-1 + x / / \\ \/ sign\-1 + x / / / \\ \/ sign\-1 + x / / / \\ \/ sign\-1 + x / / /
- -------------- - -------------- + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ - --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
3/2 5/2 / _________\ | _________| / _________\ | _________| / _________\ | _________| 3 3 2 2 2 2 2 2 3 3 2 2
5 / 1 \ 7 / 1 \ | / 2 | | / 2 | | / 2 | | / 2 | | / 2 | | / 2 | / _________\ | _________| / _________\ | _________| / _________\ | _________| / _________\ | _________| / _________\ | _________| / _________\ | _________| / _________\ | _________| _________ / _________\ | _________|
x *|1 - --| x *|1 - --| \x + \/ -1 + x /*|x + \/ -1 + x | \x + \/ -1 + x /*|x + \/ -1 + x | \x + \/ -1 + x /*|x + \/ -1 + x | | / 2 | | / 2 | | / 2 | | / 2 | | / 2 | | / 2 | | / 2 | | / 2 | | / 2 | | / 2 | | / 2 | | / 2 | | / 2 | | / 2 | / 2 | / 2 | | / 2 |
| 2| | 2| \x + \/ -1 + x / *|x + \/ -1 + x | \x + \/ -1 + x / *|x + \/ -1 + x | \x + \/ -1 + x / *|x + \/ -1 + x | \x + \/ -1 + x / *|x + \/ -1 + x | \x + \/ -1 + x / *|x + \/ -1 + x | \x + \/ -1 + x / *|x + \/ -1 + x | \x + \/ -1 + x / *|x + \/ -1 + x | \/ -1 + x *\x + \/ -1 + x / *|x + \/ -1 + x |
\ x / \ x /
$$\frac{\left(x \sin^{2}{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)} \operatorname{sign}{\left(x^{2} - 1 \right)} + \left(x + \sqrt{\frac{x^{2} - 1}{\operatorname{sign}{\left(x^{2} - 1 \right)}}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)}\right) \left(\frac{x \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)} \operatorname{sign}{\left(x^{2} - 1 \right)}}{\sqrt{\frac{x^{2} - 1}{\operatorname{sign}{\left(x^{2} - 1 \right)}}}} + 1\right)\right) \frac{d^{2}}{d x^{2}} \operatorname{sign}{\left(x + \sqrt{x^{2} - 1} \right)}}{\left(x + \sqrt{x^{2} - 1}\right) \left|{x + \sqrt{x^{2} - 1}}\right|} + \frac{2 \left(4 x^{2} \sin^{2}{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)} \delta\left(x^{2} - 1\right) - \left(\frac{x \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)} \operatorname{sign}{\left(x^{2} - 1 \right)}}{\sqrt{\frac{x^{2} - 1}{\operatorname{sign}{\left(x^{2} - 1 \right)}}}} + 1\right) \left(\frac{x \sqrt{\frac{x^{2} - 1}{\operatorname{sign}{\left(x^{2} - 1 \right)}}} \left(\frac{2 \left(x^{2} - 1\right) \delta\left(x^{2} - 1\right)}{\operatorname{sign}{\left(x^{2} - 1 \right)}} - 1\right) \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)}}{x^{2} - 1} - 1\right) + \sin^{2}{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)} \operatorname{sign}{\left(x^{2} - 1 \right)} + \frac{\left(x + \sqrt{\frac{x^{2} - 1}{\operatorname{sign}{\left(x^{2} - 1 \right)}}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)}\right) \left(4 x^{2} \delta\left(x^{2} - 1\right) + \frac{x^{2} \left(\frac{2 \left(x^{2} - 1\right) \delta\left(x^{2} - 1\right)}{\operatorname{sign}{\left(x^{2} - 1 \right)}} - 1\right) \operatorname{sign}{\left(x^{2} - 1 \right)}}{x^{2} - 1} + \operatorname{sign}{\left(x^{2} - 1 \right)}\right) \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)}}{\sqrt{\frac{x^{2} - 1}{\operatorname{sign}{\left(x^{2} - 1 \right)}}}}\right) \frac{d}{d x} \operatorname{sign}{\left(x + \sqrt{x^{2} - 1} \right)}}{\left(x + \sqrt{x^{2} - 1}\right) \left|{x + \sqrt{x^{2} - 1}}\right|} + \frac{\left(8 x^{3} \sin^{2}{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)} \delta^{\left( 1 \right)}\left( x^{2} - 1 \right) + 12 x \sin^{2}{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)} \delta\left(x^{2} - 1\right) + \frac{x \left(x + \sqrt{\frac{x^{2} - 1}{\operatorname{sign}{\left(x^{2} - 1 \right)}}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)}\right) \left(8 x^{2} \delta^{\left( 1 \right)}\left( x^{2} - 1 \right) + \frac{12 x^{2} \left(\frac{2 \left(x^{2} - 1\right) \delta\left(x^{2} - 1\right)}{\operatorname{sign}{\left(x^{2} - 1 \right)}} - 1\right) \delta\left(x^{2} - 1\right)}{x^{2} - 1} + \frac{x^{2} \left(\frac{2 \left(x^{2} - 1\right) \delta\left(x^{2} - 1\right)}{\operatorname{sign}{\left(x^{2} - 1 \right)}} - 1\right)^{2} \operatorname{sign}{\left(x^{2} - 1 \right)}}{\left(x^{2} - 1\right)^{2}} - \frac{2 x^{2} \left(\frac{2 \left(x^{2} - 1\right) \delta\left(x^{2} - 1\right)}{\operatorname{sign}{\left(x^{2} - 1 \right)}} - 1\right) \operatorname{sign}{\left(x^{2} - 1 \right)}}{\left(x^{2} - 1\right)^{2}} + 12 \delta\left(x^{2} - 1\right) + \frac{2 \left(\frac{2 \left(x^{2} - 1\right) \delta\left(x^{2} - 1\right)}{\operatorname{sign}{\left(x^{2} - 1 \right)}} - 1\right) \operatorname{sign}{\left(x^{2} - 1 \right)}}{x^{2} - 1} + \frac{\left(- \frac{16 x^{2} \left(x^{2} - 1\right) \left(\delta\left(x^{2} - 1\right)\right)^{2}}{\operatorname{sign}^{2}{\left(x^{2} - 1 \right)}} + \frac{4 x^{2} \left(x^{2} - 1\right) \delta^{\left( 1 \right)}\left( x^{2} - 1 \right)}{\operatorname{sign}{\left(x^{2} - 1 \right)}} + \frac{8 x^{2} \delta\left(x^{2} - 1\right)}{\operatorname{sign}{\left(x^{2} - 1 \right)}} + \frac{2 \left(x^{2} - 1\right) \delta\left(x^{2} - 1\right)}{\operatorname{sign}{\left(x^{2} - 1 \right)}} - 1\right) \operatorname{sign}{\left(x^{2} - 1 \right)}}{x^{2} - 1}\right) \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)}}{\sqrt{\frac{x^{2} - 1}{\operatorname{sign}{\left(x^{2} - 1 \right)}}}} - \frac{\sqrt{\frac{x^{2} - 1}{\operatorname{sign}{\left(x^{2} - 1 \right)}}} \left(\frac{x \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)} \operatorname{sign}{\left(x^{2} - 1 \right)}}{\sqrt{\frac{x^{2} - 1}{\operatorname{sign}{\left(x^{2} - 1 \right)}}}} + 1\right) \left(- \frac{16 x^{2} \left(x^{2} - 1\right) \left(\delta\left(x^{2} - 1\right)\right)^{2}}{\operatorname{sign}^{2}{\left(x^{2} - 1 \right)}} + \frac{4 x^{2} \left(x^{2} - 1\right) \delta^{\left( 1 \right)}\left( x^{2} - 1 \right)}{\operatorname{sign}{\left(x^{2} - 1 \right)}} + \frac{4 x^{2} \left(\frac{2 \left(x^{2} - 1\right) \delta\left(x^{2} - 1\right)}{\operatorname{sign}{\left(x^{2} - 1 \right)}} - 1\right) \delta\left(x^{2} - 1\right)}{\operatorname{sign}{\left(x^{2} - 1 \right)}} + \frac{8 x^{2} \delta\left(x^{2} - 1\right)}{\operatorname{sign}{\left(x^{2} - 1 \right)}} - \frac{x^{2} \left(\frac{2 \left(x^{2} - 1\right) \delta\left(x^{2} - 1\right)}{\operatorname{sign}{\left(x^{2} - 1 \right)}} - 1\right)^{2}}{x^{2} - 1} - \frac{2 x^{2} \left(\frac{2 \left(x^{2} - 1\right) \delta\left(x^{2} - 1\right)}{\operatorname{sign}{\left(x^{2} - 1 \right)}} - 1\right)}{x^{2} - 1} + \frac{2 \left(x^{2} - 1\right) \delta\left(x^{2} - 1\right)}{\operatorname{sign}{\left(x^{2} - 1 \right)}} - 1\right) \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)}}{x^{2} - 1} - \frac{2 \left(\frac{x \sqrt{\frac{x^{2} - 1}{\operatorname{sign}{\left(x^{2} - 1 \right)}}} \left(\frac{2 \left(x^{2} - 1\right) \delta\left(x^{2} - 1\right)}{\operatorname{sign}{\left(x^{2} - 1 \right)}} - 1\right) \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)}}{x^{2} - 1} - 1\right) \left(4 x^{2} \delta\left(x^{2} - 1\right) + \frac{x^{2} \left(\frac{2 \left(x^{2} - 1\right) \delta\left(x^{2} - 1\right)}{\operatorname{sign}{\left(x^{2} - 1 \right)}} - 1\right) \operatorname{sign}{\left(x^{2} - 1 \right)}}{x^{2} - 1} + \operatorname{sign}{\left(x^{2} - 1 \right)}\right) \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)}}{\sqrt{\frac{x^{2} - 1}{\operatorname{sign}{\left(x^{2} - 1 \right)}}}}\right) \operatorname{sign}{\left(x + \sqrt{x^{2} - 1} \right)}}{\left(x + \sqrt{x^{2} - 1}\right) \left|{x + \sqrt{x^{2} - 1}}\right|} - \frac{2 \left(\frac{x}{\sqrt{x^{2} - 1}} + 1\right) \left(x \sin^{2}{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)} \operatorname{sign}{\left(x^{2} - 1 \right)} + \left(x + \sqrt{\frac{x^{2} - 1}{\operatorname{sign}{\left(x^{2} - 1 \right)}}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)}\right) \left(\frac{x \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)} \operatorname{sign}{\left(x^{2} - 1 \right)}}{\sqrt{\frac{x^{2} - 1}{\operatorname{sign}{\left(x^{2} - 1 \right)}}}} + 1\right)\right) \frac{d}{d x} \operatorname{sign}{\left(x + \sqrt{x^{2} - 1} \right)}}{\left(x + \sqrt{x^{2} - 1}\right)^{2} \left|{x + \sqrt{x^{2} - 1}}\right|} - \frac{2 \left(\frac{x}{\sqrt{x^{2} - 1}} + 1\right) \left(4 x^{2} \sin^{2}{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)} \delta\left(x^{2} - 1\right) - \left(\frac{x \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)} \operatorname{sign}{\left(x^{2} - 1 \right)}}{\sqrt{\frac{x^{2} - 1}{\operatorname{sign}{\left(x^{2} - 1 \right)}}}} + 1\right) \left(\frac{x \sqrt{\frac{x^{2} - 1}{\operatorname{sign}{\left(x^{2} - 1 \right)}}} \left(\frac{2 \left(x^{2} - 1\right) \delta\left(x^{2} - 1\right)}{\operatorname{sign}{\left(x^{2} - 1 \right)}} - 1\right) \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)}}{x^{2} - 1} - 1\right) + \sin^{2}{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)} \operatorname{sign}{\left(x^{2} - 1 \right)} + \frac{\left(x + \sqrt{\frac{x^{2} - 1}{\operatorname{sign}{\left(x^{2} - 1 \right)}}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)}\right) \left(4 x^{2} \delta\left(x^{2} - 1\right) + \frac{x^{2} \left(\frac{2 \left(x^{2} - 1\right) \delta\left(x^{2} - 1\right)}{\operatorname{sign}{\left(x^{2} - 1 \right)}} - 1\right) \operatorname{sign}{\left(x^{2} - 1 \right)}}{x^{2} - 1} + \operatorname{sign}{\left(x^{2} - 1 \right)}\right) \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)}}{\sqrt{\frac{x^{2} - 1}{\operatorname{sign}{\left(x^{2} - 1 \right)}}}}\right) \operatorname{sign}{\left(x + \sqrt{x^{2} - 1} \right)}}{\left(x + \sqrt{x^{2} - 1}\right)^{2} \left|{x + \sqrt{x^{2} - 1}}\right|} - \frac{3 \left(x \sin^{2}{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)} \operatorname{sign}{\left(x^{2} - 1 \right)} + \left(x + \sqrt{\frac{x^{2} - 1}{\operatorname{sign}{\left(x^{2} - 1 \right)}}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)}\right) \left(\frac{x \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)} \operatorname{sign}{\left(x^{2} - 1 \right)}}{\sqrt{\frac{x^{2} - 1}{\operatorname{sign}{\left(x^{2} - 1 \right)}}}} + 1\right)\right)^{2} \operatorname{sign}{\left(x + \sqrt{x^{2} - 1} \right)} \frac{d}{d x} \operatorname{sign}{\left(x + \sqrt{x^{2} - 1} \right)}}{\left(x + \sqrt{x^{2} - 1}\right)^{2} \left|{x + \sqrt{x^{2} - 1}}\right|^{2}} - \frac{3 \left(x \sin^{2}{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)} \operatorname{sign}{\left(x^{2} - 1 \right)} + \left(x + \sqrt{\frac{x^{2} - 1}{\operatorname{sign}{\left(x^{2} - 1 \right)}}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)}\right) \left(\frac{x \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)} \operatorname{sign}{\left(x^{2} - 1 \right)}}{\sqrt{\frac{x^{2} - 1}{\operatorname{sign}{\left(x^{2} - 1 \right)}}}} + 1\right)\right) \left(4 x^{2} \sin^{2}{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)} \delta\left(x^{2} - 1\right) - \left(\frac{x \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)} \operatorname{sign}{\left(x^{2} - 1 \right)}}{\sqrt{\frac{x^{2} - 1}{\operatorname{sign}{\left(x^{2} - 1 \right)}}}} + 1\right) \left(\frac{x \sqrt{\frac{x^{2} - 1}{\operatorname{sign}{\left(x^{2} - 1 \right)}}} \left(\frac{2 \left(x^{2} - 1\right) \delta\left(x^{2} - 1\right)}{\operatorname{sign}{\left(x^{2} - 1 \right)}} - 1\right) \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)}}{x^{2} - 1} - 1\right) + \sin^{2}{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)} \operatorname{sign}{\left(x^{2} - 1 \right)} + \frac{\left(x + \sqrt{\frac{x^{2} - 1}{\operatorname{sign}{\left(x^{2} - 1 \right)}}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)}\right) \left(4 x^{2} \delta\left(x^{2} - 1\right) + \frac{x^{2} \left(\frac{2 \left(x^{2} - 1\right) \delta\left(x^{2} - 1\right)}{\operatorname{sign}{\left(x^{2} - 1 \right)}} - 1\right) \operatorname{sign}{\left(x^{2} - 1 \right)}}{x^{2} - 1} + \operatorname{sign}{\left(x^{2} - 1 \right)}\right) \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)}}{\sqrt{\frac{x^{2} - 1}{\operatorname{sign}{\left(x^{2} - 1 \right)}}}}\right) \operatorname{sign}^{2}{\left(x + \sqrt{x^{2} - 1} \right)}}{\left(x + \sqrt{x^{2} - 1}\right)^{2} \left|{x + \sqrt{x^{2} - 1}}\right|^{2}} + \frac{\left(\frac{x^{2}}{x^{2} - 1} - 1\right) \left(x \sin^{2}{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)} \operatorname{sign}{\left(x^{2} - 1 \right)} + \left(x + \sqrt{\frac{x^{2} - 1}{\operatorname{sign}{\left(x^{2} - 1 \right)}}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)}\right) \left(\frac{x \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)} \operatorname{sign}{\left(x^{2} - 1 \right)}}{\sqrt{\frac{x^{2} - 1}{\operatorname{sign}{\left(x^{2} - 1 \right)}}}} + 1\right)\right) \operatorname{sign}{\left(x + \sqrt{x^{2} - 1} \right)}}{\left(x + \sqrt{x^{2} - 1}\right)^{2} \sqrt{x^{2} - 1} \left|{x + \sqrt{x^{2} - 1}}\right|} + \frac{2 \left(\frac{x}{\sqrt{x^{2} - 1}} + 1\right)^{2} \left(x \sin^{2}{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)} \operatorname{sign}{\left(x^{2} - 1 \right)} + \left(x + \sqrt{\frac{x^{2} - 1}{\operatorname{sign}{\left(x^{2} - 1 \right)}}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)}\right) \left(\frac{x \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)} \operatorname{sign}{\left(x^{2} - 1 \right)}}{\sqrt{\frac{x^{2} - 1}{\operatorname{sign}{\left(x^{2} - 1 \right)}}}} + 1\right)\right) \operatorname{sign}{\left(x + \sqrt{x^{2} - 1} \right)}}{\left(x + \sqrt{x^{2} - 1}\right)^{3} \left|{x + \sqrt{x^{2} - 1}}\right|} + \frac{3 \left(\frac{x}{\sqrt{x^{2} - 1}} + 1\right) \left(x \sin^{2}{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)} \operatorname{sign}{\left(x^{2} - 1 \right)} + \left(x + \sqrt{\frac{x^{2} - 1}{\operatorname{sign}{\left(x^{2} - 1 \right)}}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)}\right) \left(\frac{x \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)} \operatorname{sign}{\left(x^{2} - 1 \right)}}{\sqrt{\frac{x^{2} - 1}{\operatorname{sign}{\left(x^{2} - 1 \right)}}}} + 1\right)\right)^{2} \operatorname{sign}^{2}{\left(x + \sqrt{x^{2} - 1} \right)}}{\left(x + \sqrt{x^{2} - 1}\right)^{3} \left|{x + \sqrt{x^{2} - 1}}\right|^{2}} + \frac{2 \left(x \sin^{2}{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)} \operatorname{sign}{\left(x^{2} - 1 \right)} + \left(x + \sqrt{\frac{x^{2} - 1}{\operatorname{sign}{\left(x^{2} - 1 \right)}}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)}\right) \left(\frac{x \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,x^{2} - 1 \right)}}{2} \right)} \operatorname{sign}{\left(x^{2} - 1 \right)}}{\sqrt{\frac{x^{2} - 1}{\operatorname{sign}{\left(x^{2} - 1 \right)}}}} + 1\right)\right)^{3} \operatorname{sign}^{3}{\left(x + \sqrt{x^{2} - 1} \right)}}{\left(x + \sqrt{x^{2} - 1}\right)^{3} \left|{x + \sqrt{x^{2} - 1}}\right|^{3}} - \frac{4}{x^{5} \left(1 - \frac{1}{x^{2}}\right)^{\frac{3}{2}}} - \frac{3}{x^{7} \left(1 - \frac{1}{x^{2}}\right)^{\frac{5}{2}}}$$
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