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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Reescribimos las funciones para diferenciar:
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Se aplica la regla de la derivada parcial:
y .
Para calcular :
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La derivada del seno es igual al coseno:
Para calcular :
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La derivada del coseno es igual a menos el seno:
Ahora aplicamos la regla de la derivada de una divesión:
Como resultado de:
Simplificamos:
Respuesta:
/ tan(x)\ / tan(x)\ / tan(x) \
\e / / 2 \ \e / |e / 2 \ tan(x) |
x *\1 + tan (x)/ + x *|------- + \1 + tan (x)/*e *log(x)|*tan(x)
\ x /
$$x^{e^{\tan{\left(x \right)}}} \left(\left(\tan^{2}{\left(x \right)} + 1\right) e^{\tan{\left(x \right)}} \log{\left(x \right)} + \frac{e^{\tan{\left(x \right)}}}{x}\right) \tan{\left(x \right)} + x^{e^{\tan{\left(x \right)}}} \left(\tan^{2}{\left(x \right)} + 1\right)$$
/ tan(x)\ / / 2 2 / 2 \ \ \
\e / | / 2 \ | 1 / 2 \ /1 / 2 \ \ tan(x) 2*\1 + tan (x)/ / 2 \ | tan(x) / 2 \ /1 / 2 \ \ tan(x)|
x *|2*\1 + tan (x)/*tan(x) + |- -- + \1 + tan (x)/ *log(x) + |- + \1 + tan (x)/*log(x)| *e + --------------- + 2*\1 + tan (x)/*log(x)*tan(x)|*e *tan(x) + 2*\1 + tan (x)/*|- + \1 + tan (x)/*log(x)|*e |
| | 2 \x / x | \x / |
\ \ x / /
$$x^{e^{\tan{\left(x \right)}}} \left(2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{1}{x}\right) \left(\tan^{2}{\left(x \right)} + 1\right) e^{\tan{\left(x \right)}} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \left(\left(\left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{1}{x}\right)^{2} e^{\tan{\left(x \right)}} + \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \log{\left(x \right)} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} \tan{\left(x \right)} + \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)}{x} - \frac{1}{x^{2}}\right) e^{\tan{\left(x \right)}} \tan{\left(x \right)}\right)$$
/ / 2 \ \
/ tan(x)\ | | 3 3 / 2 \ 2 / 2 \ / 2 / 2 \ \ / 2 \ 2 | / 2 2 / 2 \ \ |
\e / | / 2 \ / 2 \ |2 / 2 \ /1 / 2 \ \ 2*tan(x) 3*\1 + tan (x)/ / 2 \ 3*\1 + tan (x)/ /1 / 2 \ \ | 1 / 2 \ 2*\1 + tan (x)/ / 2 \ | tan(x) 2 / 2 \ 6*\1 + tan (x)/*tan(x) / 2 \ | tan(x) / 2 \ | 1 / 2 \ /1 / 2 \ \ tan(x) 2*\1 + tan (x)/ / 2 \ | tan(x) / 2 \ /1 / 2 \ \ tan(x) |
x *|2*\1 + tan (x)/*\1 + 3*tan (x)/ + |-- + \1 + tan (x)/ *log(x) + |- + \1 + tan (x)/*log(x)| *e - --------------- + 2*\1 + tan (x)/ *log(x) + ---------------- + 3*|- + \1 + tan (x)/*log(x)|*|- -- + \1 + tan (x)/ *log(x) + --------------- + 2*\1 + tan (x)/*log(x)*tan(x)|*e + 4*tan (x)*\1 + tan (x)/*log(x) + ---------------------- + 6*\1 + tan (x)/ *log(x)*tan(x)|*e *tan(x) + 3*\1 + tan (x)/*|- -- + \1 + tan (x)/ *log(x) + |- + \1 + tan (x)/*log(x)| *e + --------------- + 2*\1 + tan (x)/*log(x)*tan(x)|*e + 6*\1 + tan (x)/*|- + \1 + tan (x)/*log(x)|*e *tan(x)|
| | 3 \x / 2 x \x / | 2 x | x | | 2 \x / x | \x / |
\ \x x \ x / / \ x / /
$$x^{e^{\tan{\left(x \right)}}} \left(6 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{1}{x}\right) \left(\tan^{2}{\left(x \right)} + 1\right) e^{\tan{\left(x \right)}} \tan{\left(x \right)} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) + 3 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\left(\left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{1}{x}\right)^{2} e^{\tan{\left(x \right)}} + \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \log{\left(x \right)} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} \tan{\left(x \right)} + \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)}{x} - \frac{1}{x^{2}}\right) e^{\tan{\left(x \right)}} + \left(\left(\left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{1}{x}\right)^{3} e^{2 \tan{\left(x \right)}} + 3 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{1}{x}\right) \left(\left(\tan^{2}{\left(x \right)} + 1\right)^{2} \log{\left(x \right)} + 2 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} \tan{\left(x \right)} + \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)}{x} - \frac{1}{x^{2}}\right) e^{\tan{\left(x \right)}} + \left(\tan^{2}{\left(x \right)} + 1\right)^{3} \log{\left(x \right)} + 6 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \log{\left(x \right)} \tan{\left(x \right)} + 2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \log{\left(x \right)} + 4 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} \tan^{2}{\left(x \right)} + \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{x} + \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}}{x} - \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)}{x^{2}} + \frac{2}{x^{3}}\right) e^{\tan{\left(x \right)}} \tan{\left(x \right)}\right)$$