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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Sustituimos .
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Derivado es .
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Luego se aplica una cadena de reglas. Multiplicamos por :
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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 la secuencia de reglas:
Como resultado de la secuencia de reglas:
Simplificamos:
Respuesta:
5 / 2 \
6*log (tan(x))*\1 + tan (x)/
----------------------------
tan(x)
$$\frac{6 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(\tan{\left(x \right)} \right)}^{5}}{\tan{\left(x \right)}}$$
/ / 2 \ / 2 \ \
4 / 2 \ | 5*\1 + tan (x)/ \1 + tan (x)/*log(tan(x))|
6*log (tan(x))*\1 + tan (x)/*|2*log(tan(x)) + --------------- - -------------------------|
| 2 2 |
\ tan (x) tan (x) /
$$6 \left(\tan^{2}{\left(x \right)} + 1\right) \left(- \frac{\left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(\tan{\left(x \right)} \right)}}{\tan^{2}{\left(x \right)}} + \frac{5 \left(\tan^{2}{\left(x \right)} + 1\right)}{\tan^{2}{\left(x \right)}} + 2 \log{\left(\tan{\left(x \right)} \right)}\right) \log{\left(\tan{\left(x \right)} \right)}^{4}$$
/ 2 2 2 \
| / 2 \ / 2 \ 2 / 2 \ / 2 \ 2 / 2 \ |
3 / 2 \ | 2 20*\1 + tan (x)/ 15*\1 + tan (x)/ *log(tan(x)) 4*log (tan(x))*\1 + tan (x)/ 2*\1 + tan (x)/ *log (tan(x)) 30*\1 + tan (x)/*log(tan(x))|
6*log (tan(x))*\1 + tan (x)/*|4*log (tan(x))*tan(x) + ----------------- - ----------------------------- - ---------------------------- + ----------------------------- + ----------------------------|
| 3 3 tan(x) 3 tan(x) |
\ tan (x) tan (x) tan (x) /
$$6 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \log{\left(\tan{\left(x \right)} \right)}^{2}}{\tan^{3}{\left(x \right)}} - \frac{15 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \log{\left(\tan{\left(x \right)} \right)}}{\tan^{3}{\left(x \right)}} + \frac{20 \left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\tan^{3}{\left(x \right)}} - \frac{4 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(\tan{\left(x \right)} \right)}^{2}}{\tan{\left(x \right)}} + \frac{30 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(\tan{\left(x \right)} \right)}}{\tan{\left(x \right)}} + 4 \log{\left(\tan{\left(x \right)} \right)}^{2} \tan{\left(x \right)}\right) \log{\left(\tan{\left(x \right)} \right)}^{3}$$
/ 2 2 2 \
| / 2 \ / 2 \ 2 / 2 \ / 2 \ 2 / 2 \ |
3 / 2 \ | 2 20*\1 + tan (x)/ 15*\1 + tan (x)/ *log(tan(x)) 4*log (tan(x))*\1 + tan (x)/ 2*\1 + tan (x)/ *log (tan(x)) 30*\1 + tan (x)/*log(tan(x))|
6*log (tan(x))*\1 + tan (x)/*|4*log (tan(x))*tan(x) + ----------------- - ----------------------------- - ---------------------------- + ----------------------------- + ----------------------------|
| 3 3 tan(x) 3 tan(x) |
\ tan (x) tan (x) tan (x) /
$$6 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \log{\left(\tan{\left(x \right)} \right)}^{2}}{\tan^{3}{\left(x \right)}} - \frac{15 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \log{\left(\tan{\left(x \right)} \right)}}{\tan^{3}{\left(x \right)}} + \frac{20 \left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\tan^{3}{\left(x \right)}} - \frac{4 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(\tan{\left(x \right)} \right)}^{2}}{\tan{\left(x \right)}} + \frac{30 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(\tan{\left(x \right)} \right)}}{\tan{\left(x \right)}} + 4 \log{\left(\tan{\left(x \right)} \right)}^{2} \tan{\left(x \right)}\right) \log{\left(\tan{\left(x \right)} \right)}^{3}$$