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:
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Derivada de x^-8
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Derivada de x^2/lnx
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Derivada de √x+2
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Derivada de (t^(2)+1)÷(t^(1÷2)-1)
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Expresiones idénticas
- x*-ln(abs(uno /(x+ uno)- uno / dos +sqrt((uno /(x+ uno)- uno / dos)^ dos + tres / cuatro)))exp(-x)
- x multiplicar por menos ln(abs(1 dividir por (x más 1) menos 1 dividir por 2 más raíz cuadrada de ((1 dividir por (x más 1) menos 1 dividir por 2) al cuadrado más 3 dividir por 4))) exponente de ( menos x)
- x multiplicar por menos ln(abs(uno dividir por (x más uno) menos uno dividir por dos más raíz cuadrada de ((uno dividir por (x más uno) menos uno dividir por dos) en el grado dos más tres dividir por cuatro))) exponente de ( menos x)
- x*-ln(abs(1/(x+1)-1/2+√((1/(x+1)-1/2)^2+3/4)))exp(-x)
- x*-ln(abs(1/(x+1)-1/2+sqrt((1/(x+1)-1/2)2+3/4)))exp(-x)
- x*-lnabs1/x+1-1/2+sqrt1/x+1-1/22+3/4exp-x
- x*-ln(abs(1/(x+1)-1/2+sqrt((1/(x+1)-1/2)²+3/4)))exp(-x)
- x*-ln(abs(1/(x+1)-1/2+sqrt((1/(x+1)-1/2) en el grado 2+3/4)))exp(-x)
- x-ln(abs(1/(x+1)-1/2+sqrt((1/(x+1)-1/2)^2+3/4)))exp(-x)
- x-ln(abs(1/(x+1)-1/2+sqrt((1/(x+1)-1/2)2+3/4)))exp(-x)
- x-lnabs1/x+1-1/2+sqrt1/x+1-1/22+3/4exp-x
- x-lnabs1/x+1-1/2+sqrt1/x+1-1/2^2+3/4exp-x
- x*-ln(abs(1 dividir por (x+1)-1 dividir por 2+sqrt((1 dividir por (x+1)-1 dividir por 2)^2+3 dividir por 4)))exp(-x)
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Expresiones semejantes
- x*+ln(abs(1/(x+1)-1/2+sqrt((1/(x+1)-1/2)^2+3/4)))exp(-x)
- x*-ln(abs(1/(x-1)-1/2+sqrt((1/(x+1)-1/2)^2+3/4)))exp(-x)
- x*-ln(abs(1/(x+1)-1/2+sqrt((1/(x+1)-1/2)^2-3/4)))exp(-x)
- x*-ln(abs(1/(x+1)-1/2+sqrt((1/(x-1)-1/2)^2+3/4)))exp(-x)
- x*-ln(abs(1/(x+1)-1/2+sqrt((1/(x+1)+1/2)^2+3/4)))exp(-x)
- x*-ln(abs(1/(x+1)-1/2-sqrt((1/(x+1)-1/2)^2+3/4)))exp(-x)
- x*-ln(abs(1/(x+1)-1/2+sqrt((1/(x+1)-1/2)^2+3/4)))exp(x)
- x*-ln(abs(1/(x+1)+1/2+sqrt((1/(x+1)-1/2)^2+3/4)))exp(-x)
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Expresiones con funciones
- ln
- ln(x/5)
- ln(×)
- ln(1+x²)
- ln(x+10)^10
- ln(x+1/x)
- Raíz cuadrada sqrt
- sqrt(x^2+16)+3x
- sqrt(5*x+1)
- sqrt(sinx^3)
- sqrt(log(6*x-1))
- sqrt(x)-1/sqrt(x)
Derivada de x*-ln(abs(1/(x+1)-1/2+sqrt((1/(x+1)-1/2)^2+3/4)))exp(-x)
Solución
Ha introducido
[src]
/ /| __________________|\\ | || / 2 ||| | || 1 1 / / 1 1\ 3 ||| -x x*|-log||----- - - + / |----- - -| + - |||*e \ \|x + 1 2 \/ \x + 1 2/ 4 |//
$$x \left(- \log{\left(\left|{\left(- \frac{1}{2} + \frac{1}{x + 1}\right) + \sqrt{\left(- \frac{1}{2} + \frac{1}{x + 1}\right)^{2} + \frac{3}{4}}}\right| \right)}\right) e^{- x}$$
(x*(-log(Abs(1/(x + 1) - 1/2 + sqrt((1/(x + 1) - 1/2)^2 + 3/4)))))*exp(-x)
Primera derivada
[src]
/ // _______________________ / / 2\\\ _______________________ / / 2\\ \ \ | || / 2 | | 3 / 1 1 \ ||| / 2 | | 3 / 1 1 \ || | | | || / / 2\ |atan2|0, - + |- - + -----| ||| / _______________________ / / 2\\\ / / 2\ |atan2|0, - + |- - + -----| || | | | || / |3 / 1 1 \ | / 1 1 \ | \ 4 \ 2 1 + x/ /|| | / 2 | | 3 / 1 1 \ ||| / |3 / 1 1 \ | 2| \ 4 \ 2 1 + x/ /| / 1 1 \| / ____________________\| | || 4 / |- + |- - + -----| | *|- - + -----|*cos|----------------------------|| | / / 2\ |atan2|0, - + |- - + -----| ||| / |- + |- - + -----| | *sin |----------------------------|*|- - + -----|| | / 2 || | || 1 \/ \4 \ 2 1 + x/ / \ 2 1 + x/ \ 2 /| | 1 1 / |3 / 1 1 \ | | \ 4 \ 2 1 + x/ /|| \/ \4 \ 2 1 + x/ / \ 2 / \ 2 1 + x/| | 1 1 / 3 / 1 1 \ || | x*||- -------- - -----------------------------------------------------------------------------|*|- - + ----- + 4 / |- + |- - + -----| | *cos|----------------------------|| - ------------------------------------------------------------------------------|*sign|- - + ----- + / - + |- - + -----| || | || 2 / 2\ | \ 2 1 + x \/ \4 \ 2 1 + x/ / \ 2 // / 2\ | \ 2 1 + x \/ 4 \ 2 1 + x/ /| | /| __________________|\ || (1 + x) 2 |3 / 1 1 \ | | 2 |3 / 1 1 \ | | | /| __________________|\ | || / 2 || || (1 + x) *|- + |- - + -----| | | (1 + x) *|- + |- - + -----| | | | || / 2 || | || 1 1 / / 1 1\ 3 || \\ \4 \ 2 1 + x/ / / \4 \ 2 1 + x/ / / | -x -x || 1 1 / / 1 1\ 3 || |- log||----- - - + / |----- - -| + - || - ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|*e + x*e *log||----- - - + / |----- - -| + - || | \|x + 1 2 \/ \x + 1 2/ 4 |/ / ____________________\ | __________________| | \|x + 1 2 \/ \x + 1 2/ 4 |/ | | / 2 | | / 2 | | | | 1 1 / 3 / 1 1 \ | | 1 1 / / 1 1\ 3 | | | |- - + ----- + / - + |- - + -----| |*|----- - - + / |----- - -| + - | | \ \ 2 1 + x \/ 4 \ 2 1 + x/ / |x + 1 2 \/ \x + 1 2/ 4 | /
$$x e^{- x} \log{\left(\left|{\left(- \frac{1}{2} + \frac{1}{x + 1}\right) + \sqrt{\left(- \frac{1}{2} + \frac{1}{x + 1}\right)^{2} + \frac{3}{4}}}\right| \right)} + \left(- \frac{x \left(- \frac{\left(- \frac{1}{2} + \frac{1}{x + 1}\right) \sqrt{\left(\left(- \frac{1}{2} + \frac{1}{x + 1}\right)^{2} + \frac{3}{4}\right)^{2}} \sin^{2}{\left(\frac{\operatorname{atan_{2}}{\left(0,\left(- \frac{1}{2} + \frac{1}{x + 1}\right)^{2} + \frac{3}{4} \right)}}{2} \right)}}{\left(x + 1\right)^{2} \left(\left(- \frac{1}{2} + \frac{1}{x + 1}\right)^{2} + \frac{3}{4}\right)} + \left(- \frac{\left(- \frac{1}{2} + \frac{1}{x + 1}\right) \sqrt[4]{\left(\left(- \frac{1}{2} + \frac{1}{x + 1}\right)^{2} + \frac{3}{4}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,\left(- \frac{1}{2} + \frac{1}{x + 1}\right)^{2} + \frac{3}{4} \right)}}{2} \right)}}{\left(x + 1\right)^{2} \left(\left(- \frac{1}{2} + \frac{1}{x + 1}\right)^{2} + \frac{3}{4}\right)} - \frac{1}{\left(x + 1\right)^{2}}\right) \left(\sqrt[4]{\left(\left(- \frac{1}{2} + \frac{1}{x + 1}\right)^{2} + \frac{3}{4}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,\left(- \frac{1}{2} + \frac{1}{x + 1}\right)^{2} + \frac{3}{4} \right)}}{2} \right)} - \frac{1}{2} + \frac{1}{x + 1}\right)\right) \operatorname{sign}{\left(\sqrt{\left(- \frac{1}{2} + \frac{1}{x + 1}\right)^{2} + \frac{3}{4}} - \frac{1}{2} + \frac{1}{x + 1} \right)}}{\left(\sqrt{\left(- \frac{1}{2} + \frac{1}{x + 1}\right)^{2} + \frac{3}{4}} - \frac{1}{2} + \frac{1}{x + 1}\right) \left|{\left(- \frac{1}{2} + \frac{1}{x + 1}\right) + \sqrt{\left(- \frac{1}{2} + \frac{1}{x + 1}\right)^{2} + \frac{3}{4}}}\right|} - \log{\left(\left|{\left(- \frac{1}{2} + \frac{1}{x + 1}\right) + \sqrt{\left(- \frac{1}{2} + \frac{1}{x + 1}\right)^{2} + \frac{3}{4}}}\right| \right)}\right) e^{- x}$$
Segunda derivada
[src]
/ / / 2 \ \ \ | | | / / / 2\\\ | | | | | | | | | / 2 \ ||| | | | | | | | _____________________ | | 3 + |1 - -----| ||| | | | | | | | / 2 | | \ 1 + x/ ||| | | | | | | | / / 2\ |atan2|0, ----------------||| | | | | | | / / / 2\\ / / 2\\ / / 2\\\ | / | / 2 \ | / 2 \ | \ 4 /|| / / 2\\ / / 2\\| 2 // / / 2\\\ / / 2\\ \ | | | | | | | | / 2 \ || | | / 2 \ || | | / 2 \ ||| | 4 / |3 + |1 - -----| | *|1 - -----|*cos|--------------------------|| | | / 2 \ || | | / 2 \ ||| // / / 2\\\ / / 2\\ \ || | | / 2 \ ||| | | / 2 \ || | | | | | |/ / / 2\\\ | _____________________ | | 3 + |1 - -----| || _____________________ | | 3 + |1 - -----| || _____________________ | | 3 + |1 - -----| ||| | \/ \ \ 1 + x/ / \ 1 + x/ \ 2 /| _____________________ | | 3 + |1 - -----| || _____________________ | | 3 + |1 - -----| ||| || | | / 2 \ ||| | | / 2 \ || | || _____________________ | | 3 + |1 - -----| ||| / / / 2\\\ _____________________ | | 3 + |1 - -----| || | | | | | || | | / 2 \ ||| | / 2 | | \ 1 + x/ || / 2 | | \ 1 + x/ || / 2 | | \ 1 + x/ ||| |-1 + -----------------------------------------------------------------------| / 2 | | \ 1 + x/ || / 2 | | \ 1 + x/ ||| / __________________\ || _____________________ | | 3 + |1 - -----| ||| / / / 2\\\ _____________________ | | 3 + |1 - -----| || | || / 2 | | \ 1 + x/ ||| | | | / 2 \ ||| / 2 | | \ 1 + x/ || | / __________________\| | | | || _____________________ | | 3 + |1 - -----| ||| | / / 2\ |atan2|0, ----------------|| / / 2\ |atan2|0, ----------------|| / / 2\ 2 |atan2|0, ----------------||| | 2 | / / 2\ |atan2|0, ----------------|| / / 2\ |atan2|0, ----------------||| | / 2 | || / 2 | | \ 1 + x/ ||| | | | / 2 \ ||| / 2 | | \ 1 + x/ || | / __________________\ || / / 2\ |atan2|0, ----------------||| | _____________________ | | 3 + |1 - -----| ||| / / 2\ |atan2|0, ----------------|| | | / 2 || | | | || / 2 | | \ 1 + x/ ||| | / | / 2 \ | / 2 \ | \ 4 /| / | / 2 \ | | \ 4 /| / | / 2 \ | / 2 \ | \ 4 /|| | / 2 \ | / | / 2 \ | 2| \ 4 /| / 2 \ / | / 2 \ | 2| \ 4 /|| | / / 2 \ | || / / 2\ |atan2|0, ----------------||| | _____________________ | | 3 + |1 - -----| ||| / / 2\ |atan2|0, ----------------|| | | / 2 | / 2 \ || / | / 2 \ | / 2 \ | \ 4 /|| | / 2 | | \ 1 + x/ ||| / | / 2 \ | 2| \ 4 /| / 2 \| | / / 2 \ || | | | // / / 2\\\ / / 2\\ \ || / / 2\ |atan2|0, ----------------||| | 4 / |3 + |1 - -----| | *|1 - -----|*cos|--------------------------| 4 / |3 + |1 - -----| | *cos|--------------------------| 4 / |3 + |1 - -----| | *|1 - -----| *cos|--------------------------|| | 3 + |1 - -----| | / |3 + |1 - -----| | *sin |--------------------------|*|1 - -----| / |3 + |1 - -----| | *sin |--------------------------|| | / 3 + |1 - -----| | || / | / 2 \ | / 2 \ | \ 4 /|| | / 2 | | \ 1 + x/ ||| / | / 2 \ | 2| \ 4 /| / 2 \| | / / 2 \ | | 1 - ----- | || 4 / |3 + |1 - -----| | *|1 - -----|*cos|--------------------------|| | / / 2\ |atan2|0, ----------------||| / |3 + |1 - -----| | *sin |--------------------------|*|1 - -----|| | / 3 + |1 - -----| || // / / 2\\\ / / 2\\ \ | | | || | | / 2 \ ||| | | / 2 \ || | || 2 / | / 2 \ | | \ 4 /|| | \/ \ \ 1 + x/ / \ 1 + x/ \ 2 / \/ \ \ 1 + x/ / \ 2 / \/ \ \ 1 + x/ / \ 1 + x/ \ 2 /| \ \ 1 + x/ / \/ \ \ 1 + x/ / \ 2 / \ 1 + x/ \/ \ \ 1 + x/ / \ 2 /| | 1 1 \/ \ 1 + x/ | || 4 / |3 + |1 - -----| | *|1 - -----|*cos|--------------------------|| | / / 2\ |atan2|0, ----------------||| / |3 + |1 - -----| | *sin |--------------------------|*|1 - -----|| | / 3 + |1 - -----| | | 1 + x | || \/ \ \ 1 + x/ / \ 1 + x/ \ 2 /| | 2 / | / 2 \ | | \ 4 /|| \/ \ \ 1 + x/ / \ 2 / \ 1 + x/| | 1 1 \/ \ 1 + x/ || || | | / 2 \ ||| | | / 2 \ || | | | | || _____________________ | | 3 + |1 - -----| ||| / / / 2\\\ _____________________ | | 3 + |1 - -----| || | 2*||-1 + ----- + 4 / |3 + |1 - -----| | *cos|--------------------------||*|-1 + ----------------------------------------------------------------------- - ----------------------------------------------------------- + ------------------------------------------------------------------------| - ------------------------------------------------------------------------------- + ------------------------------------------------------------------------ - ------------------------------------------------------------|*sign|- - + ----- + -----------------------| || \/ \ \ 1 + x/ / \ 1 + x/ \ 2 /| | 2 / | / 2 \ | | \ 4 /|| \/ \ \ 1 + x/ / \ 2 / \ 1 + x/| 2| 1 1 \/ \ 1 + x/ | 2*|-1 + -----------------------|*||-1 + -----------------------------------------------------------------------|*|-1 + ----- + 4 / |3 + |1 - -----| | *cos|--------------------------|| + ------------------------------------------------------------------------|*sign|- - + ----- + -----------------------|| || _____________________ | | 3 + |1 - -----| ||| / / / 2\\\ _____________________ | | 3 + |1 - -----| || | | | | || / 2 | | \ 1 + x/ ||| | | | / 2 \ ||| / 2 | | \ 1 + x/ || | / / __________________\\ |\ 1 + x \/ \ \ 1 + x/ / \ 2 // | 2 / 2\ 2 | 1 + x 2 / 2\ | \ 2 1 + x 2 / ||-1 + -----------------------------------------------------------------------|*|-1 + ----- + 4 / |3 + |1 - -----| | *cos|--------------------------|| + ------------------------------------------------------------------------| *sign |- - + ----- + -----------------------| | __________________| || 2 | \ 1 + x \/ \ \ 1 + x/ / \ 2 // 2 | \ 2 1 + x 2 /| || / 2 | | \ 1 + x/ ||| | | | / 2 \ ||| / 2 | | \ 1 + x/ || | / __________________\ // / / 2\\\ / / 2\\ \ | | | || / / 2\ |atan2|0, ----------------||| | _____________________ | | 3 + |1 - -----| ||| / / 2\ |atan2|0, ----------------|| | | | / 2 || | | / 2 \ | / 2 \ | / 2\ | / 2 \ | / 2 \ | | || 2 | \ 1 + x \/ \ \ 1 + x/ / \ 2 // 2 | \ 2 1 + x 2 / | / 2 | || / 2 \ | / 2 \ | | || / / 2\ |atan2|0, ----------------||| | _____________________ | | 3 + |1 - -----| ||| / / 2\ |atan2|0, ----------------|| | | / 2 | || | | / 2 \ ||| | | / 2 \ || | | | | || / | / 2 \ | / 2 \ | \ 4 /|| | / 2 | | \ 1 + x/ ||| / | / 2 \ | 2| \ 4 /| / 2 \| | | / / 2 \ || | | 3 + |1 - -----| (1 + x)*|3 + |1 - -----| | | / 2 \ | | 3 + |1 - -----| (1 + x)*|3 + |1 - -----| | | || / 2 \ | / 2 \ | | / / 2 \ | || 3 + |1 - -----| | 3 + |1 - -----| | | || / | / 2 \ | / 2 \ | \ 4 /|| | / 2 | | \ 1 + x/ ||| / | / 2 \ | 2| \ 4 /| / 2 \| | / / 2 \ | || ______________________ | | 3 + |-1 + -----| ||| / / / 2\\\ _____________________ | | 3 + |1 - -----| || | | | | || 4 / |3 + |1 - -----| | *|1 - -----|*cos|--------------------------|| | / / 2\ |atan2|0, ----------------||| / |3 + |1 - -----| | *sin |--------------------------|*|1 - -----|| | | / 3 + |1 - -----| || | | \ 1 + x/ \ \ 1 + x/ / (1 + x)*|3 + |1 - -----| | | \ 1 + x/ \ \ 1 + x/ / | || 3 + |1 - -----| | 3 + |1 - -----| | | / 3 + |1 - -----| | \\ \ 1 + x/ / \ 1 + x/ / | || 4 / |3 + |1 - -----| | *|1 - -----|*cos|--------------------------|| | / / 2\ |atan2|0, ----------------||| / |3 + |1 - -----| | *sin |--------------------------|*|1 - -----|| | / 3 + |1 - -----| | || / 2 | | \ 1 + x/ ||| | | | / 2 \ ||| / 2 | | \ 1 + x/ || | / __________________\| | | || \/ \ \ 1 + x/ / \ 1 + x/ \ 2 /| | 2 / | / 2 \ | | \ 4 /|| \/ \ \ 1 + x/ / \ 2 / \ 1 + x/| d | | 1 1 \/ \ 1 + x/ || \ \ \ \ 1 + x/ / / / \\ \ 1 + x/ / \ 1 + x/ / \ \/ \ 1 + x/ / | || \/ \ \ 1 + x/ / \ 1 + x/ \ 2 /| | 2 / | / 2 \ | | \ 4 /|| \/ \ \ 1 + x/ / \ 2 / \ 1 + x/| | 1 1 \/ \ 1 + x/ | || / / 2\ |atan2|0, -----------------||| | _____________________ | | 3 + |1 - -----| ||| / / 2\ |atan2|0, ----------------|| | | / 2 || | x*|- ||-1 + -----------------------------------------------------------------------|*|-1 + ----- + 4 / |3 + |1 - -----| | *cos|--------------------------|| + ------------------------------------------------------------------------|*--|sign|- - + ----- + -----------------------|| + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ + -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| - 2*||-1 + -----------------------------------------------------------------------|*|-1 + ----- + 4 / |3 + |1 - -----| | *cos|--------------------------|| + ------------------------------------------------------------------------|*sign|- - + ----- + -----------------------| || / | / 2 \ | / 2 \ | \ 4 /|| | / 2 | | \ 1 + x/ ||| / | / 2 \ | 2| \ 4 /| / 2 \| | / / 2 \ || | | || 2 | \ 1 + x \/ \ \ 1 + x/ / \ 2 // 2 | dx\ \ 2 1 + x 2 // 1 + x | __________________| / __________________ \ | || 2 | \ 1 + x \/ \ \ 1 + x/ / \ 2 // 2 | \ 2 1 + x 2 / || 4 / |3 + |-1 + -----| | *|-1 + -----|*cos|---------------------------|| | / / 2\ |atan2|0, ----------------||| / |3 + |1 - -----| | *sin |--------------------------|*|1 - -----|| | / 3 + |1 - -----| || | | || / 2 \ | / 2 \ | | / 2 | | / 2 | | || / 2 \ | / 2 \ | || \/ \ \ 1 + x/ / \ 1 + x/ \ 2 /| | 2 / | / 2 \ | | \ 4 /|| \/ \ \ 1 + x/ / \ 2 / \ 1 + x/| | 1 1 \/ \ 1 + x/ || | /| ___________________|\ /| __________________|\ | || 3 + |1 - -----| | 3 + |1 - -----| | / __________________ \ | / / 2 \ | 2 | / / 2 \ 2 | | || 3 + |1 - -----| | 3 + |1 - -----| | 2*x*||1 + --------------------------------------------------------------------------|*|-1 + ----- + 4 / |3 + |1 - -----| | *cos|--------------------------|| - ------------------------------------------------------------------------|*sign|- - + ----- + -----------------------|| | || / 2 || || / 2 || | \\ \ 1 + x/ / \ 1 + x/ / | / 2 | | / 3 + |1 - -----| | (1 + x) *|-1 + / 3 + |1 - -----| + -----| | \\ \ 1 + x/ / \ 1 + x/ / || 2 | \ 1 + x \/ \ \ 1 + x/ / \ 2 // 2 | \ 2 1 + x 2 /| | || / / 2 \ || || / / 2 \ || | 2 | / / 2 \ 2 | | 1 1 \/ \ 1 + x/ | \ \/ \ 1 + x/ 1 + x/ | || / 2 \ | / 2 \ | | | || / 3 + |-1 + -----| || || / 3 + |1 - -----| || | (1 + x) *|-1 + / 3 + |1 - -----| + -----|*|- - + ----- + -----------------------| | || 3 + |-1 + -----| | 3 + |1 - -----| | | | || 1 1 \/ \ 1 + x/ || || 1 1 \/ \ 1 + x/ || \ \ \/ \ 1 + x/ 1 + x/ | 2 1 + x 2 | / \\ \ 1 + x/ / \ 1 + x/ / | -x |2*log||- - + ----- + ------------------------|| - x*log||- - + ----- + -----------------------|| + --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|*e | \| 2 1 + x 2 |/ \| 2 1 + x 2 |/ | __________________| | ___________________| | | | / 2 | | / 2 | | | / __________________ \ | / / 2 \ | / __________________ \ | / / 2 \ | | | | / 2 | | / 3 + |1 - -----| | | / 2 | | / 3 + |-1 + -----| | | | 2 | / / 2 \ 2 | | 1 1 \/ \ 1 + x/ | 2 | / / 2 \ 2 | | 1 1 \/ \ 1 + x/ | | | (1 + x) *|-1 + / 3 + |1 - -----| + -----|*|- - + ----- + -----------------------| (1 + x) *|-1 + / 3 + |1 - -----| + -----|*|- - + ----- + ------------------------| | \ \ \/ \ 1 + x/ 1 + x/ | 2 1 + x 2 | \ \/ \ 1 + x/ 1 + x/ | 2 1 + x 2 | /
$$\left(- x \log{\left(\left|{\frac{\sqrt{\left(1 - \frac{2}{x + 1}\right)^{2} + 3}}{2} - \frac{1}{2} + \frac{1}{x + 1}}\right| \right)} - \frac{2 x \left(- \frac{\left(1 - \frac{2}{x + 1}\right) \sqrt{\left(\left(1 - \frac{2}{x + 1}\right)^{2} + 3\right)^{2}} \sin^{2}{\left(\frac{\operatorname{atan_{2}}{\left(0,\frac{\left(1 - \frac{2}{x + 1}\right)^{2} + 3}{4} \right)}}{2} \right)}}{\left(1 - \frac{2}{x + 1}\right)^{2} + 3} + \left(\frac{\left(-1 + \frac{2}{x + 1}\right) \sqrt[4]{\left(\left(-1 + \frac{2}{x + 1}\right)^{2} + 3\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,\frac{\left(-1 + \frac{2}{x + 1}\right)^{2} + 3}{4} \right)}}{2} \right)}}{\left(-1 + \frac{2}{x + 1}\right)^{2} + 3} + 1\right) \left(\sqrt[4]{\left(\left(1 - \frac{2}{x + 1}\right)^{2} + 3\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,\frac{\left(1 - \frac{2}{x + 1}\right)^{2} + 3}{4} \right)}}{2} \right)} - 1 + \frac{2}{x + 1}\right)\right) \operatorname{sign}{\left(\frac{\sqrt{\left(1 - \frac{2}{x + 1}\right)^{2} + 3}}{2} - \frac{1}{2} + \frac{1}{x + 1} \right)}}{\left(x + 1\right)^{2} \left(\sqrt{\left(1 - \frac{2}{x + 1}\right)^{2} + 3} - 1 + \frac{2}{x + 1}\right) \left|{\frac{\sqrt{\left(-1 + \frac{2}{x + 1}\right)^{2} + 3}}{2} - \frac{1}{2} + \frac{1}{x + 1}}\right|} + 2 \log{\left(\left|{\frac{\sqrt{\left(-1 + \frac{2}{x + 1}\right)^{2} + 3}}{2} - \frac{1}{2} + \frac{1}{x + 1}}\right| \right)} + \frac{x \left(- \left(\frac{\left(1 - \frac{2}{x + 1}\right) \sqrt{\left(\left(1 - \frac{2}{x + 1}\right)^{2} + 3\right)^{2}} \sin^{2}{\left(\frac{\operatorname{atan_{2}}{\left(0,\frac{\left(1 - \frac{2}{x + 1}\right)^{2} + 3}{4} \right)}}{2} \right)}}{\left(1 - \frac{2}{x + 1}\right)^{2} + 3} + \left(\frac{\left(1 - \frac{2}{x + 1}\right) \sqrt[4]{\left(\left(1 - \frac{2}{x + 1}\right)^{2} + 3\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,\frac{\left(1 - \frac{2}{x + 1}\right)^{2} + 3}{4} \right)}}{2} \right)}}{\left(1 - \frac{2}{x + 1}\right)^{2} + 3} - 1\right) \left(\sqrt[4]{\left(\left(1 - \frac{2}{x + 1}\right)^{2} + 3\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,\frac{\left(1 - \frac{2}{x + 1}\right)^{2} + 3}{4} \right)}}{2} \right)} - 1 + \frac{2}{x + 1}\right)\right) \frac{d}{d x} \operatorname{sign}{\left(\frac{\sqrt{\left(1 - \frac{2}{x + 1}\right)^{2} + 3}}{2} - \frac{1}{2} + \frac{1}{x + 1} \right)} + \frac{2 \left(\frac{\left(1 - \frac{2}{x + 1}\right) \sqrt{\left(\left(1 - \frac{2}{x + 1}\right)^{2} + 3\right)^{2}} \sin^{2}{\left(\frac{\operatorname{atan_{2}}{\left(0,\frac{\left(1 - \frac{2}{x + 1}\right)^{2} + 3}{4} \right)}}{2} \right)}}{\left(1 - \frac{2}{x + 1}\right)^{2} + 3} + \left(\sqrt[4]{\left(\left(1 - \frac{2}{x + 1}\right)^{2} + 3\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,\frac{\left(1 - \frac{2}{x + 1}\right)^{2} + 3}{4} \right)}}{2} \right)} - 1 + \frac{2}{x + 1}\right) \left(\frac{\left(1 - \frac{2}{x + 1}\right)^{2} \sqrt[4]{\left(\left(1 - \frac{2}{x + 1}\right)^{2} + 3\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,\frac{\left(1 - \frac{2}{x + 1}\right)^{2} + 3}{4} \right)}}{2} \right)}}{\left(x + 1\right) \left(\left(1 - \frac{2}{x + 1}\right)^{2} + 3\right)^{2}} + \frac{\left(1 - \frac{2}{x + 1}\right) \sqrt[4]{\left(\left(1 - \frac{2}{x + 1}\right)^{2} + 3\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,\frac{\left(1 - \frac{2}{x + 1}\right)^{2} + 3}{4} \right)}}{2} \right)}}{\left(1 - \frac{2}{x + 1}\right)^{2} + 3} - 1 - \frac{\sqrt[4]{\left(\left(1 - \frac{2}{x + 1}\right)^{2} + 3\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,\frac{\left(1 - \frac{2}{x + 1}\right)^{2} + 3}{4} \right)}}{2} \right)}}{\left(x + 1\right) \left(\left(1 - \frac{2}{x + 1}\right)^{2} + 3\right)}\right) - \frac{\left(\frac{\left(1 - \frac{2}{x + 1}\right) \sqrt[4]{\left(\left(1 - \frac{2}{x + 1}\right)^{2} + 3\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,\frac{\left(1 - \frac{2}{x + 1}\right)^{2} + 3}{4} \right)}}{2} \right)}}{\left(1 - \frac{2}{x + 1}\right)^{2} + 3} - 1\right)^{2}}{x + 1} - \frac{\sqrt{\left(\left(1 - \frac{2}{x + 1}\right)^{2} + 3\right)^{2}} \sin^{2}{\left(\frac{\operatorname{atan_{2}}{\left(0,\frac{\left(1 - \frac{2}{x + 1}\right)^{2} + 3}{4} \right)}}{2} \right)}}{\left(x + 1\right) \left(\left(1 - \frac{2}{x + 1}\right)^{2} + 3\right)}\right) \operatorname{sign}{\left(\frac{\sqrt{\left(1 - \frac{2}{x + 1}\right)^{2} + 3}}{2} - \frac{1}{2} + \frac{1}{x + 1} \right)}}{x + 1} + \frac{2 \left(\frac{1 - \frac{2}{x + 1}}{\sqrt{\left(1 - \frac{2}{x + 1}\right)^{2} + 3}} - 1\right) \left(\frac{\left(1 - \frac{2}{x + 1}\right) \sqrt{\left(\left(1 - \frac{2}{x + 1}\right)^{2} + 3\right)^{2}} \sin^{2}{\left(\frac{\operatorname{atan_{2}}{\left(0,\frac{\left(1 - \frac{2}{x + 1}\right)^{2} + 3}{4} \right)}}{2} \right)}}{\left(1 - \frac{2}{x + 1}\right)^{2} + 3} + \left(\frac{\left(1 - \frac{2}{x + 1}\right) \sqrt[4]{\left(\left(1 - \frac{2}{x + 1}\right)^{2} + 3\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,\frac{\left(1 - \frac{2}{x + 1}\right)^{2} + 3}{4} \right)}}{2} \right)}}{\left(1 - \frac{2}{x + 1}\right)^{2} + 3} - 1\right) \left(\sqrt[4]{\left(\left(1 - \frac{2}{x + 1}\right)^{2} + 3\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,\frac{\left(1 - \frac{2}{x + 1}\right)^{2} + 3}{4} \right)}}{2} \right)} - 1 + \frac{2}{x + 1}\right)\right) \operatorname{sign}{\left(\frac{\sqrt{\left(1 - \frac{2}{x + 1}\right)^{2} + 3}}{2} - \frac{1}{2} + \frac{1}{x + 1} \right)}}{\left(x + 1\right)^{2} \left(\sqrt{\left(1 - \frac{2}{x + 1}\right)^{2} + 3} - 1 + \frac{2}{x + 1}\right)} + \frac{\left(\frac{\left(1 - \frac{2}{x + 1}\right) \sqrt{\left(\left(1 - \frac{2}{x + 1}\right)^{2} + 3\right)^{2}} \sin^{2}{\left(\frac{\operatorname{atan_{2}}{\left(0,\frac{\left(1 - \frac{2}{x + 1}\right)^{2} + 3}{4} \right)}}{2} \right)}}{\left(1 - \frac{2}{x + 1}\right)^{2} + 3} + \left(\frac{\left(1 - \frac{2}{x + 1}\right) \sqrt[4]{\left(\left(1 - \frac{2}{x + 1}\right)^{2} + 3\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,\frac{\left(1 - \frac{2}{x + 1}\right)^{2} + 3}{4} \right)}}{2} \right)}}{\left(1 - \frac{2}{x + 1}\right)^{2} + 3} - 1\right) \left(\sqrt[4]{\left(\left(1 - \frac{2}{x + 1}\right)^{2} + 3\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,\frac{\left(1 - \frac{2}{x + 1}\right)^{2} + 3}{4} \right)}}{2} \right)} - 1 + \frac{2}{x + 1}\right)\right)^{2} \operatorname{sign}^{2}{\left(\frac{\sqrt{\left(1 - \frac{2}{x + 1}\right)^{2} + 3}}{2} - \frac{1}{2} + \frac{1}{x + 1} \right)}}{\left(x + 1\right)^{2} \left(\sqrt{\left(1 - \frac{2}{x + 1}\right)^{2} + 3} - 1 + \frac{2}{x + 1}\right) \left|{\frac{\sqrt{\left(1 - \frac{2}{x + 1}\right)^{2} + 3}}{2} - \frac{1}{2} + \frac{1}{x + 1}}\right|}\right) - 2 \left(\frac{\left(1 - \frac{2}{x + 1}\right) \sqrt{\left(\left(1 - \frac{2}{x + 1}\right)^{2} + 3\right)^{2}} \sin^{2}{\left(\frac{\operatorname{atan_{2}}{\left(0,\frac{\left(1 - \frac{2}{x + 1}\right)^{2} + 3}{4} \right)}}{2} \right)}}{\left(1 - \frac{2}{x + 1}\right)^{2} + 3} + \left(\frac{\left(1 - \frac{2}{x + 1}\right) \sqrt[4]{\left(\left(1 - \frac{2}{x + 1}\right)^{2} + 3\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,\frac{\left(1 - \frac{2}{x + 1}\right)^{2} + 3}{4} \right)}}{2} \right)}}{\left(1 - \frac{2}{x + 1}\right)^{2} + 3} - 1\right) \left(\sqrt[4]{\left(\left(1 - \frac{2}{x + 1}\right)^{2} + 3\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(0,\frac{\left(1 - \frac{2}{x + 1}\right)^{2} + 3}{4} \right)}}{2} \right)} - 1 + \frac{2}{x + 1}\right)\right) \operatorname{sign}{\left(\frac{\sqrt{\left(1 - \frac{2}{x + 1}\right)^{2} + 3}}{2} - \frac{1}{2} + \frac{1}{x + 1} \right)}}{\left(x + 1\right)^{2} \left(\sqrt{\left(1 - \frac{2}{x + 1}\right)^{2} + 3} - 1 + \frac{2}{x + 1}\right) \left|{\frac{\sqrt{\left(1 - \frac{2}{x + 1}\right)^{2} + 3}}{2} - \frac{1}{2} + \frac{1}{x + 1}}\right|}\right) e^{- x}$$