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 e^x*sin(x)
-
Derivada de x*e^(1/x)
- Derivada de x!
-
Derivada de e^y
-
Expresiones idénticas
- y=sqrt((x^ cinco + uno)/(xlnx))
- y es igual a raíz cuadrada de ((x en el grado 5 más 1) dividir por (xlnx))
- y es igual a raíz cuadrada de ((x en el grado cinco más uno) dividir por (xlnx))
- y=√((x^5+1)/(xlnx))
- y=sqrt((x5+1)/(xlnx))
- y=sqrtx5+1/xlnx
- y=sqrt((x⁵+1)/(xlnx))
- y=sqrtx^5+1/xlnx
- y=sqrt((x^5+1) dividir por (xlnx))
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Expresiones semejantes
- y=sqrt((x^5-1)/(xlnx))
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Expresiones con funciones
- Raíz cuadrada sqrt
- sqrt(x-1)/(sqrt(x^2)-x-1)
- sqrt(x)/(x+1)
- sqrt(x)+1
- sqrt(ln(x))
- sqrt(x)*cos^5*(3x)
- xlnx
- xlnx+xln2
- xlnx-x+1
- xlnx+arcsin(sqrt(x))
- xlnx+arcsinsqrtx
- xlnx-exp^-0.5x^2
Derivada de y=sqrt((x^5+1)/(xlnx))
Solución
Ha introducido
[src]
__________
/ 5
/ x + 1
/ --------
\/ x*log(x)
$$\sqrt{\frac{x^{5} + 1}{x \log{\left(x \right)}}}$$
sqrt((x^5 + 1)/((x*log(x))))
Solución detallada
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Sustituimos .
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Según el principio, aplicamos: tenemos
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Luego se aplica una cadena de reglas. Multiplicamos por :
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Se aplica la regla de la derivada parcial:
y .
Para calcular :
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diferenciamos miembro por miembro:
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La derivada de una constante es igual a cero.
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Según el principio, aplicamos: tenemos
Como resultado de:
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Para calcular :
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Se aplica la regla de la derivada de una multiplicación:
; calculamos :
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Según el principio, aplicamos: tenemos
; calculamos :
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Derivado es .
Como resultado de:
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Ahora aplicamos la regla de la derivada de una divesión:
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Como resultado de la secuencia de reglas:
-
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Simplificamos:
Respuesta:
Primera derivada
[src]
/ 4 1 \
___________________ |5*x *-------- / 5 \|
/ 1 / 5 \ | x*log(x) (-1 - log(x))*\x + 1/|
x* / --------*\x + 1/ *|------------- + ----------------------|*log(x)
\/ x*log(x) | 2 2 2 |
\ 2*x *log (x) /
-------------------------------------------------------------------------
5
x + 1
$$\frac{x \sqrt{\frac{1}{x \log{\left(x \right)}} \left(x^{5} + 1\right)} \left(\frac{5 x^{4} \frac{1}{x \log{\left(x \right)}}}{2} + \frac{\left(x^{5} + 1\right) \left(- \log{\left(x \right)} - 1\right)}{2 x^{2} \log{\left(x \right)}^{2}}\right) \log{\left(x \right)}}{x^{5} + 1}$$
Segunda derivada
[src]
/ / / 5\ / 1 \\ \
| / / 5\ \ / / 5\ \ | \1 + x /*|1 + ------|| / / 5\ \ |
| | 3 \1 + x /*(1 + log(x))| 5 | 3 \1 + x /*(1 + log(x))| 2 | 3 \ log(x)/| | 3 \1 + x /*(1 + log(x))| |
__________ | 2*|5*x - ---------------------| 10*x *|5*x - ---------------------| x *|5*x - ---------------------|*|5*x - ---------------------| |
/ 5 | / 5 2 / 5\ / 5\ \ | 2 | | 2 | | 2 | | 2 | / 5\ |
/ 1 + x | 3 | 2 2 / 1 \ 1 + x 5*x *(1 + log(x)) 2*\1 + x /*(1 + log(x)) 2*\1 + x /*(1 + log(x))| \ x *log(x) / \ x *log(x) / \ x / \ x *log(x) / 2*\1 + x /*(1 + log(x))|
/ -------- *|10*x - 2*x*|- 20*x + 5*x *|1 + ------| + --------- + ----------------- - ----------------------- - -----------------------| + -------------------------------- - ------------------------------------ + ---------------------------------------------------------------- - -----------------------|
\/ x*log(x) | | \ log(x)/ 3 log(x) 3 3 2 | log(x) 5 5 2 |
\ \ x *log(x) x *log(x) x *log (x) / 1 + x 1 + x x *log(x) /
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
/ 5\
4*\1 + x /
$$\frac{\sqrt{\frac{x^{5} + 1}{x \log{\left(x \right)}}} \left(- \frac{10 x^{5} \left(5 x^{3} - \frac{\left(x^{5} + 1\right) \left(\log{\left(x \right)} + 1\right)}{x^{2} \log{\left(x \right)}}\right)}{x^{5} + 1} + 10 x^{3} + \frac{x^{2} \left(5 x^{3} - \frac{\left(1 + \frac{1}{\log{\left(x \right)}}\right) \left(x^{5} + 1\right)}{x^{2}}\right) \left(5 x^{3} - \frac{\left(x^{5} + 1\right) \left(\log{\left(x \right)} + 1\right)}{x^{2} \log{\left(x \right)}}\right)}{x^{5} + 1} - 2 x \left(5 x^{2} \left(1 + \frac{1}{\log{\left(x \right)}}\right) + \frac{5 x^{2} \left(\log{\left(x \right)} + 1\right)}{\log{\left(x \right)}} - 20 x^{2} - \frac{2 \left(x^{5} + 1\right) \left(\log{\left(x \right)} + 1\right)}{x^{3} \log{\left(x \right)}} - \frac{2 \left(x^{5} + 1\right) \left(\log{\left(x \right)} + 1\right)}{x^{3} \log{\left(x \right)}^{2}} + \frac{x^{5} + 1}{x^{3} \log{\left(x \right)}}\right) + \frac{2 \left(5 x^{3} - \frac{\left(x^{5} + 1\right) \left(\log{\left(x \right)} + 1\right)}{x^{2} \log{\left(x \right)}}\right)}{\log{\left(x \right)}} - \frac{2 \left(x^{5} + 1\right) \left(\log{\left(x \right)} + 1\right)}{x^{2} \log{\left(x \right)}}\right)}{4 \left(x^{5} + 1\right)}$$
Tercera derivada
[src]
/ / / 5\ / 2 3 \\ \
| / / 5\ / 1 \\ / / 5\ / 1 \\ / / 5\ / 1 \\ | \1 + x /*|2 + ------- + ------|| / / 5\ / 1 \\ / / 5\ / 1 \\ |
| / / 5\ / 5\ / 5\ / 5\ / 5\ \ 5 2 / 5\ / 5\ / 5\ / / 5\ \ / 5 2 / 5\ / 5\ \ / / 5\ \ | \1 + x /*|1 + ------|| / / 5\ \ / / 5\ \ | \1 + x /*|1 + ------|| / / 5\ \ / / 5\ \ / 5 5 \ | \1 + x /*|1 + ------|| / 5 2 / 5\ / 5\ \ / / 5\ \ | 2 | 2 log(x)|| / 5 5 \ / 5 2 / 5\ / 5\ \ / / 5\ \ / 5 5 \ | \1 + x /*|1 + ------|| / / 5\ \ / / 5\ \ / 5 5 \ | \1 + x /*|1 + ------|| / / 5\ \ / 5 5 \|
| | 30*x / 1 \ / 2 3 \ 4*\1 + x / 5*\1 + x / 20*x*(1 + log(x)) 10*\1 + x /*(1 + log(x)) 6*\1 + x /*(1 + log(x)) 6*\1 + x /*(1 + log(x))| 2 2 / 1 \ 1 + x 5*x *(1 + log(x)) 2*\1 + x /*(1 + log(x)) 2*\1 + x /*(1 + log(x)) 3 \1 + x /*(1 + log(x)) 4 | 3 \1 + x /*(1 + log(x))| 5 | 2 2 / 1 \ 1 + x 5*x *(1 + log(x)) 2*\1 + x /*(1 + log(x)) 2*\1 + x /*(1 + log(x))| 9 | 3 \1 + x /*(1 + log(x))| | 3 \ log(x)/| | 3 \1 + x /*(1 + log(x))| 4 | 3 \1 + x /*(1 + log(x))| 6 | 3 \ log(x)/| | 3 \1 + x /*(1 + log(x))| | 3 \1 + x /*(1 + log(x))| | 3 1 + x 1 + x | 2 | 3 \ log(x)/| | 2 2 / 1 \ 1 + x 5*x *(1 + log(x)) 2*\1 + x /*(1 + log(x)) 2*\1 + x /*(1 + log(x))| 2 | 3 \1 + x /*(1 + log(x))| | 2 5*x 2 / 1 \ \ log (x) /| 2 | 3 1 + x 1 + x | | 2 2 / 1 \ 1 + x 5*x *(1 + log(x)) 2*\1 + x /*(1 + log(x)) 2*\1 + x /*(1 + log(x))| 6 | 3 \1 + x /*(1 + log(x))| | 3 1 + x 1 + x | | 3 \ log(x)/| | 3 \1 + x /*(1 + log(x))| | 3 \1 + x /*(1 + log(x))| | 3 1 + x 1 + x | 3 | 3 \ log(x)/| | 3 \1 + x /*(1 + log(x))| | 3 1 + x 1 + x ||
__________ | x*|40*x - ------ - 20*x*|1 + ------| + 5*x*|2 + ------- + ------| + ---------- + ---------- + ----------------- - ------------------------ - ----------------------- - -----------------------| - 20*x + 5*x *|1 + ------| + --------- + ----------------- - ----------------------- - ----------------------- 5*x - --------------------- 15*x *|5*x - ---------------------| 5*x *|- 20*x + 5*x *|1 + ------| + --------- + ----------------- - ----------------------- - -----------------------| 25*x *|5*x - ---------------------| x*|5*x - ---------------------|*|5*x - ---------------------| 5*x *|5*x - ---------------------| 5*x *|5*x - ---------------------|*|5*x - ---------------------| x*|5*x - ---------------------|*|- 5*x + ------ + ---------| x *|5*x - ---------------------|*|- 20*x + 5*x *|1 + ------| + --------- + ----------------- - ----------------------- - -----------------------| x *|5*x - ---------------------|*|- 15*x + ------ + 5*x *|1 + ------| - -------------------------------| x *|- 5*x + ------ + ---------|*|- 20*x + 5*x *|1 + ------| + --------- + ----------------- - ----------------------- - -----------------------| 5*x *|5*x - ---------------------|*|- 5*x + ------ + ---------| x*|5*x - ---------------------|*|5*x - ---------------------| x*|5*x - ---------------------|*|- 5*x + ------ + ---------| x *|5*x - ---------------------|*|5*x - ---------------------|*|- 5*x + ------ + ---------||
/ 5 | | log(x) \ log(x)/ | 2 log(x)| 4 2 4 2 4 2 4 4 3 | \ log(x)/ 3 log(x) 3 3 2 2 5 | 2 | 2 | \ log(x)/ 3 log(x) 3 3 2 | | 2 | | 2 | | 2 | | 2 | / 5\ / 5\ | 2 | | 2 | | 2 | | 2 2 | | 2 | | \ log(x)/ 3 log(x) 3 3 2 | | 2 | | log(x) \ log(x)/ 3 | | 2 2 | | \ log(x)/ 3 log(x) 3 3 2 | | 2 | | 2 2 | | 2 | | 2 | | 2 | | 2 2 | | 2 | | 2 | | 2 2 ||
/ 1 + x | 2 \ \ log (x) / x *log (x) x *log(x) log (x) x *log (x) x *log(x) x *log (x) / x *log(x) x *log(x) x *log (x) 2 / 1 \ x *log(x) 1 + x \ x *log(x) / 5*x *(1 + log(x)) \ x *log(x) x *log(x) x *log (x) / \ x *log(x) / \ x / \ x *log(x) / \ x *log(x) / 2*\1 + x /*(1 + log(x)) 2*\1 + x /*(1 + log(x)) \ x / \ x *log(x) / \ x *log(x) / \ x x *log(x)/ \ x / \ x *log(x) x *log(x) x *log (x) / \ x *log(x) / \ x / \ x x *log(x)/ \ x *log(x) x *log(x) x *log (x) / \ x *log(x) / \ x x *log(x)/ \ x / \ x *log(x) / \ x *log(x) / \ x x *log(x)/ \ x / \ x *log(x) / \ x x *log(x)/|
/ -------- *|20*x + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------------------------------- - 5*x *|1 + ------| + ---------------------------- - --------- - ------------------------------------ - ----------------- + ---------------------------------------------------------------------------------------------------------------------- + ------------------------------------ + --------------------------------------------------------------- - ----------------------------------- + ----------------------- + ----------------------- - ------------------------------------------------------------------ - -------------------------------------------------------------- - --------------------------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------- + --------------------------------------------------------------- - -------------------------------------------------------------- - ----------------------------------------------------------------------------------------------|
\/ x*log(x) | 2 log(x) \ log(x)/ 2*x*log(x) 3 5 log(x) 5 2 / 5\ / 5\ 3 3 2 2 / 5\ / 5\ / 5\ / 5\ 2 / 5\ / 5\ 2 |
| x *log(x) 1 + x 1 + x / 5\ 2*\1 + x / \1 + x /*log(x) x *log(x) x *log (x) / 5\ 4*\1 + x / 4*\1 + x / 4*\1 + x / 4*\1 + x / / 5\ 2*\1 + x /*log(x) 4*\1 + x /*log(x) / 5\ |
\ \1 + x / 2*\1 + x / 4*\1 + x / 8*\1 + x / /
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
5
1 + x
$$\frac{\sqrt{\frac{x^{5} + 1}{x \log{\left(x \right)}}} \left(\frac{25 x^{9} \left(5 x^{3} - \frac{\left(x^{5} + 1\right) \left(\log{\left(x \right)} + 1\right)}{x^{2} \log{\left(x \right)}}\right)}{\left(x^{5} + 1\right)^{2}} - \frac{5 x^{6} \left(5 x^{3} - \frac{\left(1 + \frac{1}{\log{\left(x \right)}}\right) \left(x^{5} + 1\right)}{x^{2}}\right) \left(5 x^{3} - \frac{\left(x^{5} + 1\right) \left(\log{\left(x \right)} + 1\right)}{x^{2} \log{\left(x \right)}}\right)}{2 \left(x^{5} + 1\right)^{2}} + \frac{5 x^{6} \left(5 x^{3} - \frac{\left(x^{5} + 1\right) \left(\log{\left(x \right)} + 1\right)}{x^{2} \log{\left(x \right)}}\right) \left(- 5 x^{3} + \frac{x^{5} + 1}{x^{2}} + \frac{x^{5} + 1}{x^{2} \log{\left(x \right)}}\right)}{4 \left(x^{5} + 1\right)^{2}} + \frac{5 x^{5} \left(5 x^{2} \left(1 + \frac{1}{\log{\left(x \right)}}\right) + \frac{5 x^{2} \left(\log{\left(x \right)} + 1\right)}{\log{\left(x \right)}} - 20 x^{2} - \frac{2 \left(x^{5} + 1\right) \left(\log{\left(x \right)} + 1\right)}{x^{3} \log{\left(x \right)}} - \frac{2 \left(x^{5} + 1\right) \left(\log{\left(x \right)} + 1\right)}{x^{3} \log{\left(x \right)}^{2}} + \frac{x^{5} + 1}{x^{3} \log{\left(x \right)}}\right)}{x^{5} + 1} - \frac{15 x^{4} \left(5 x^{3} - \frac{\left(x^{5} + 1\right) \left(\log{\left(x \right)} + 1\right)}{x^{2} \log{\left(x \right)}}\right)}{x^{5} + 1} - \frac{5 x^{4} \left(5 x^{3} - \frac{\left(x^{5} + 1\right) \left(\log{\left(x \right)} + 1\right)}{x^{2} \log{\left(x \right)}}\right)}{\left(x^{5} + 1\right) \log{\left(x \right)}} - \frac{x^{3} \left(5 x^{3} - \frac{\left(1 + \frac{1}{\log{\left(x \right)}}\right) \left(x^{5} + 1\right)}{x^{2}}\right) \left(5 x^{3} - \frac{\left(x^{5} + 1\right) \left(\log{\left(x \right)} + 1\right)}{x^{2} \log{\left(x \right)}}\right) \left(- 5 x^{3} + \frac{x^{5} + 1}{x^{2}} + \frac{x^{5} + 1}{x^{2} \log{\left(x \right)}}\right)}{8 \left(x^{5} + 1\right)^{2}} - 5 x^{2} \left(1 + \frac{1}{\log{\left(x \right)}}\right) - \frac{x^{2} \left(5 x^{3} - \frac{\left(1 + \frac{1}{\log{\left(x \right)}}\right) \left(x^{5} + 1\right)}{x^{2}}\right) \left(5 x^{2} \left(1 + \frac{1}{\log{\left(x \right)}}\right) + \frac{5 x^{2} \left(\log{\left(x \right)} + 1\right)}{\log{\left(x \right)}} - 20 x^{2} - \frac{2 \left(x^{5} + 1\right) \left(\log{\left(x \right)} + 1\right)}{x^{3} \log{\left(x \right)}} - \frac{2 \left(x^{5} + 1\right) \left(\log{\left(x \right)} + 1\right)}{x^{3} \log{\left(x \right)}^{2}} + \frac{x^{5} + 1}{x^{3} \log{\left(x \right)}}\right)}{4 \left(x^{5} + 1\right)} - \frac{x^{2} \left(5 x^{3} - \frac{\left(x^{5} + 1\right) \left(\log{\left(x \right)} + 1\right)}{x^{2} \log{\left(x \right)}}\right) \left(5 x^{2} \left(1 + \frac{1}{\log{\left(x \right)}}\right) - 15 x^{2} + \frac{5 x^{2}}{\log{\left(x \right)}} - \frac{\left(x^{5} + 1\right) \left(2 + \frac{3}{\log{\left(x \right)}} + \frac{2}{\log{\left(x \right)}^{2}}\right)}{x^{3}}\right)}{4 \left(x^{5} + 1\right)} - \frac{5 x^{2} \left(\log{\left(x \right)} + 1\right)}{\log{\left(x \right)}} + 20 x^{2} + \frac{x^{2} \left(- 5 x^{3} + \frac{x^{5} + 1}{x^{2}} + \frac{x^{5} + 1}{x^{2} \log{\left(x \right)}}\right) \left(5 x^{2} \left(1 + \frac{1}{\log{\left(x \right)}}\right) + \frac{5 x^{2} \left(\log{\left(x \right)} + 1\right)}{\log{\left(x \right)}} - 20 x^{2} - \frac{2 \left(x^{5} + 1\right) \left(\log{\left(x \right)} + 1\right)}{x^{3} \log{\left(x \right)}} - \frac{2 \left(x^{5} + 1\right) \left(\log{\left(x \right)} + 1\right)}{x^{3} \log{\left(x \right)}^{2}} + \frac{x^{5} + 1}{x^{3} \log{\left(x \right)}}\right)}{4 \left(x^{5} + 1\right)} + \frac{x \left(5 x^{3} - \frac{\left(1 + \frac{1}{\log{\left(x \right)}}\right) \left(x^{5} + 1\right)}{x^{2}}\right) \left(5 x^{3} - \frac{\left(x^{5} + 1\right) \left(\log{\left(x \right)} + 1\right)}{x^{2} \log{\left(x \right)}}\right)}{2 \left(x^{5} + 1\right)} + \frac{x \left(5 x^{3} - \frac{\left(1 + \frac{1}{\log{\left(x \right)}}\right) \left(x^{5} + 1\right)}{x^{2}}\right) \left(5 x^{3} - \frac{\left(x^{5} + 1\right) \left(\log{\left(x \right)} + 1\right)}{x^{2} \log{\left(x \right)}}\right)}{2 \left(x^{5} + 1\right) \log{\left(x \right)}} - \frac{x \left(5 x^{3} - \frac{\left(x^{5} + 1\right) \left(\log{\left(x \right)} + 1\right)}{x^{2} \log{\left(x \right)}}\right) \left(- 5 x^{3} + \frac{x^{5} + 1}{x^{2}} + \frac{x^{5} + 1}{x^{2} \log{\left(x \right)}}\right)}{4 \left(x^{5} + 1\right)} - \frac{x \left(5 x^{3} - \frac{\left(x^{5} + 1\right) \left(\log{\left(x \right)} + 1\right)}{x^{2} \log{\left(x \right)}}\right) \left(- 5 x^{3} + \frac{x^{5} + 1}{x^{2}} + \frac{x^{5} + 1}{x^{2} \log{\left(x \right)}}\right)}{4 \left(x^{5} + 1\right) \log{\left(x \right)}} + \frac{x \left(- 20 x \left(1 + \frac{1}{\log{\left(x \right)}}\right) + \frac{20 x \left(\log{\left(x \right)} + 1\right)}{\log{\left(x \right)}^{2}} + 5 x \left(2 + \frac{3}{\log{\left(x \right)}} + \frac{2}{\log{\left(x \right)}^{2}}\right) + 40 x - \frac{30 x}{\log{\left(x \right)}} - \frac{6 \left(x^{5} + 1\right) \left(\log{\left(x \right)} + 1\right)}{x^{4} \log{\left(x \right)}} - \frac{10 \left(x^{5} + 1\right) \left(\log{\left(x \right)} + 1\right)}{x^{4} \log{\left(x \right)}^{2}} - \frac{6 \left(x^{5} + 1\right) \left(\log{\left(x \right)} + 1\right)}{x^{4} \log{\left(x \right)}^{3}} + \frac{5 \left(x^{5} + 1\right)}{x^{4} \log{\left(x \right)}} + \frac{4 \left(x^{5} + 1\right)}{x^{4} \log{\left(x \right)}^{2}}\right)}{2} - \frac{5 x^{2} \left(1 + \frac{1}{\log{\left(x \right)}}\right) + \frac{5 x^{2} \left(\log{\left(x \right)} + 1\right)}{\log{\left(x \right)}} - 20 x^{2} - \frac{2 \left(x^{5} + 1\right) \left(\log{\left(x \right)} + 1\right)}{x^{3} \log{\left(x \right)}} - \frac{2 \left(x^{5} + 1\right) \left(\log{\left(x \right)} + 1\right)}{x^{3} \log{\left(x \right)}^{2}} + \frac{x^{5} + 1}{x^{3} \log{\left(x \right)}}}{\log{\left(x \right)}} + \frac{5 x^{3} - \frac{\left(x^{5} + 1\right) \left(\log{\left(x \right)} + 1\right)}{x^{2} \log{\left(x \right)}}}{2 x \log{\left(x \right)}} + \frac{2 \left(x^{5} + 1\right) \left(\log{\left(x \right)} + 1\right)}{x^{3} \log{\left(x \right)}} + \frac{2 \left(x^{5} + 1\right) \left(\log{\left(x \right)} + 1\right)}{x^{3} \log{\left(x \right)}^{2}} - \frac{x^{5} + 1}{x^{3} \log{\left(x \right)}}\right)}{x^{5} + 1}$$
Gráfico