Solución detallada
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Se aplica la regla de la derivada de una multiplicación:
; calculamos :
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No logro encontrar los pasos en la búsqueda de esta derivada.
Perola derivada
; calculamos :
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Según el principio, aplicamos: tenemos
Como resultado de:
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Simplificamos:
Respuesta:
x x / x \
atan (x) + x*atan (x)*|---------------- + log(atan(x))|
|/ 2\ |
\\1 + x /*atan(x) /
$$x \left(\frac{x}{\left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}} + \log{\left(\operatorname{atan}{\left(x \right)} \right)}\right) \operatorname{atan}^{x}{\left(x \right)} + \operatorname{atan}^{x}{\left(x \right)}$$
/ / 2 \ \
| | 2*x x | |
| | -2 + ------ + ----------------| |
| | 2 2 / 2\ | |
x | |/ x \ 1 + x \1 + x /*atan(x)| 2*x |
atan (x)*|2*log(atan(x)) + x*||---------------- + log(atan(x))| - ------------------------------| + ----------------|
| ||/ 2\ | / 2\ | / 2\ |
\ \\\1 + x /*atan(x) / \1 + x /*atan(x) / \1 + x /*atan(x)/
$$\left(x \left(\left(\frac{x}{\left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}} + \log{\left(\operatorname{atan}{\left(x \right)} \right)}\right)^{2} - \frac{\frac{2 x^{2}}{x^{2} + 1} + \frac{x}{\left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}} - 2}{\left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}}\right) + \frac{2 x}{\left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}} + 2 \log{\left(\operatorname{atan}{\left(x \right)} \right)}\right) \operatorname{atan}^{x}{\left(x \right)}$$
/ / 3 2 / 2 \\ / 2 \\
| | 3 8*x 2*x 6*x / x \ | 2*x x || | 2*x x ||
| | -8*x - ------- + ------ + ----------------- + ---------------- 3*|---------------- + log(atan(x))|*|-2 + ------ + ----------------|| 3*|-2 + ------ + ----------------||
| 2 | 3 atan(x) 2 / 2\ 2 / 2\ |/ 2\ | | 2 / 2\ || | 2 / 2\ ||
x | / x \ |/ x \ 1 + x \1 + x /*atan (x) \1 + x /*atan(x) \\1 + x /*atan(x) / \ 1 + x \1 + x /*atan(x)/| \ 1 + x \1 + x /*atan(x)/|
atan (x)*|3*|---------------- + log(atan(x))| + x*||---------------- + log(atan(x))| + -------------------------------------------------------------- - --------------------------------------------------------------------| - ----------------------------------|
| |/ 2\ | ||/ 2\ | 2 / 2\ | / 2\ |
| \\1 + x /*atan(x) / |\\1 + x /*atan(x) / / 2\ \1 + x /*atan(x) | \1 + x /*atan(x) |
\ \ \1 + x / *atan(x) / /
$$\left(x \left(\left(\frac{x}{\left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}} + \log{\left(\operatorname{atan}{\left(x \right)} \right)}\right)^{3} - \frac{3 \left(\frac{x}{\left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}} + \log{\left(\operatorname{atan}{\left(x \right)} \right)}\right) \left(\frac{2 x^{2}}{x^{2} + 1} + \frac{x}{\left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}} - 2\right)}{\left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}} + \frac{\frac{8 x^{3}}{x^{2} + 1} + \frac{6 x^{2}}{\left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}} - 8 x + \frac{2 x}{\left(x^{2} + 1\right) \operatorname{atan}^{2}{\left(x \right)}} - \frac{3}{\operatorname{atan}{\left(x \right)}}}{\left(x^{2} + 1\right)^{2} \operatorname{atan}{\left(x \right)}}\right) + 3 \left(\frac{x}{\left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}} + \log{\left(\operatorname{atan}{\left(x \right)} \right)}\right)^{2} - \frac{3 \left(\frac{2 x^{2}}{x^{2} + 1} + \frac{x}{\left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}} - 2\right)}{\left(x^{2} + 1\right) \operatorname{atan}{\left(x \right)}}\right) \operatorname{atan}^{x}{\left(x \right)}$$