Solución detallada
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diferenciamos miembro por miembro:
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Se aplica la regla de la derivada parcial:
y .
Para calcular :
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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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La derivada del coseno es igual a menos el seno:
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Luego se aplica una cadena de reglas. Multiplicamos por :
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Sustituimos .
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Luego se aplica una cadena de reglas. Multiplicamos por :
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La derivada del seno es igual al coseno:
Como resultado de la secuencia de reglas:
Como resultado de la secuencia de reglas:
Como resultado de la secuencia de reglas:
Para calcular :
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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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Según el principio, aplicamos: tenemos
Como resultado de la secuencia de reglas:
Ahora aplicamos la regla de la derivada de una divesión:
No logro encontrar los pasos en la búsqueda de esta derivada.
Perola derivada
Como resultado de:
Simplificamos:
Respuesta:
2 2
tan(x) /tan(x) / 2 \ \ 2/ sin(x)\ -x sin(x) / sin(x)\ -x / sin(x)\
x *|------ + \1 + tan (x)/*log(x)| - 2*x*cos \2 /*e - 2*2 *cos(x)*cos\2 /*e *log(2)*sin\2 /
\ x /
$$- 2 \cdot 2^{\sin{\left(x \right)}} e^{- x^{2}} \log{\left(2 \right)} \sin{\left(2^{\sin{\left(x \right)}} \right)} \cos{\left(2^{\sin{\left(x \right)}} \right)} \cos{\left(x \right)} - 2 x e^{- x^{2}} \cos^{2}{\left(2^{\sin{\left(x \right)}} \right)} + x^{\tan{\left(x \right)}} \left(\left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{\tan{\left(x \right)}}{x}\right)$$
2 / / 2 \ \ 2 2 2 2 2 2 2
tan(x) /tan(x) / 2 \ \ tan(x) | tan(x) 2*\1 + tan (x)/ / 2 \ | 2/ sin(x)\ -x 2 2/ sin(x)\ -x 2*sin(x) 2 2/ sin(x)\ 2 -x 2*sin(x) 2 2 2/ sin(x)\ -x sin(x) 2 2 / sin(x)\ -x / sin(x)\ sin(x) / sin(x)\ -x / sin(x)\ sin(x) / sin(x)\ -x / sin(x)\
x *|------ + \1 + tan (x)/*log(x)| + x *|- ------ + --------------- + 2*\1 + tan (x)/*log(x)*tan(x)| - 2*cos \2 /*e + 4*x *cos \2 /*e - 2*2 *cos (x)*cos \2 /*log (2)*e + 2*2 *cos (x)*log (2)*sin \2 /*e - 2*2 *cos (x)*log (2)*cos\2 /*e *sin\2 / + 2*2 *cos\2 /*e *log(2)*sin(x)*sin\2 / + 8*x*2 *cos(x)*cos\2 /*e *log(2)*sin\2 /
\ x / | 2 x |
\ x /
$$2 \cdot 2^{2 \sin{\left(x \right)}} e^{- x^{2}} \log{\left(2 \right)}^{2} \sin^{2}{\left(2^{\sin{\left(x \right)}} \right)} \cos^{2}{\left(x \right)} - 2 \cdot 2^{2 \sin{\left(x \right)}} e^{- x^{2}} \log{\left(2 \right)}^{2} \cos^{2}{\left(2^{\sin{\left(x \right)}} \right)} \cos^{2}{\left(x \right)} + 8 \cdot 2^{\sin{\left(x \right)}} x e^{- x^{2}} \log{\left(2 \right)} \sin{\left(2^{\sin{\left(x \right)}} \right)} \cos{\left(2^{\sin{\left(x \right)}} \right)} \cos{\left(x \right)} + 2 \cdot 2^{\sin{\left(x \right)}} e^{- x^{2}} \log{\left(2 \right)} \sin{\left(2^{\sin{\left(x \right)}} \right)} \sin{\left(x \right)} \cos{\left(2^{\sin{\left(x \right)}} \right)} - 2 \cdot 2^{\sin{\left(x \right)}} e^{- x^{2}} \log{\left(2 \right)}^{2} \sin{\left(2^{\sin{\left(x \right)}} \right)} \cos{\left(2^{\sin{\left(x \right)}} \right)} \cos^{2}{\left(x \right)} + 4 x^{2} e^{- x^{2}} \cos^{2}{\left(2^{\sin{\left(x \right)}} \right)} + x^{\tan{\left(x \right)}} \left(\left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{\tan{\left(x \right)}}{x}\right)^{2} + x^{\tan{\left(x \right)}} \left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} \tan{\left(x \right)} + \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right)}{x} - \frac{\tan{\left(x \right)}}{x^{2}}\right) - 2 e^{- x^{2}} \cos^{2}{\left(2^{\sin{\left(x \right)}} \right)}$$
3 / / 2 \ 2 / 2 \ \ 2 / / 2 \ \ 2 2 2 2 2 2 2 2 2 2 2 2 2 2
tan(x) /tan(x) / 2 \ \ tan(x) | 3*\1 + tan (x)/ 2*tan(x) / 2 \ 2 / 2 \ 6*\1 + tan (x)/*tan(x)| 3 2/ sin(x)\ -x tan(x) /tan(x) / 2 \ \ | tan(x) 2*\1 + tan (x)/ / 2 \ | 2/ sin(x)\ -x 2*sin(x) 3 2/ sin(x)\ 3 -x 2*sin(x) 3 3 2/ sin(x)\ -x 2*sin(x) 2 2 2/ sin(x)\ -x 2*sin(x) 2 2/ sin(x)\ -x sin(x) 3 3 / sin(x)\ -x / sin(x)\ 2*sin(x) 2/ sin(x)\ 2 -x 3*sin(x) 3 3 / sin(x)\ -x / sin(x)\ 2*sin(x) 2 2/ sin(x)\ 2 -x sin(x) / sin(x)\ -x / sin(x)\ sin(x) 2 / sin(x)\ -x / sin(x)\ sin(x) / sin(x)\ -x / sin(x)\ sin(x) 2 / sin(x)\ -x / sin(x)\ sin(x) 2 2 / sin(x)\ -x / sin(x)\
x *|------ + \1 + tan (x)/*log(x)| + x *|- --------------- + -------- + 2*\1 + tan (x)/ *log(x) + 4*tan (x)*\1 + tan (x)/*log(x) + ----------------------| - 8*x *cos \2 /*e + 3*x *|------ + \1 + tan (x)/*log(x)|*|- ------ + --------------- + 2*\1 + tan (x)/*log(x)*tan(x)| + 12*x*cos \2 /*e - 6*2 *cos (x)*cos \2 /*log (2)*e + 6*2 *cos (x)*log (2)*sin \2 /*e - 12*x*2 *cos (x)*log (2)*sin \2 /*e - 6*2 *log (2)*sin \2 /*cos(x)*e *sin(x) - 2*2 *cos (x)*log (2)*cos\2 /*e *sin\2 / + 6*2 *cos \2 /*log (2)*cos(x)*e *sin(x) + 8*2 *cos (x)*log (2)*cos\2 /*e *sin\2 / + 12*x*2 *cos (x)*cos \2 /*log (2)*e + 14*2 *cos(x)*cos\2 /*e *log(2)*sin\2 / - 24*2 *x *cos(x)*cos\2 /*e *log(2)*sin\2 / - 12*x*2 *cos\2 /*e *log(2)*sin(x)*sin\2 / + 6*2 *log (2)*cos(x)*cos\2 /*e *sin(x)*sin\2 / + 12*x*2 *cos (x)*log (2)*cos\2 /*e *sin\2 /
\ x / | 2 3 x | \ x / | 2 x |
\ x x / \ x /
$$8 \cdot 2^{3 \sin{\left(x \right)}} e^{- x^{2}} \log{\left(2 \right)}^{3} \sin{\left(2^{\sin{\left(x \right)}} \right)} \cos{\left(2^{\sin{\left(x \right)}} \right)} \cos^{3}{\left(x \right)} - 12 \cdot 2^{2 \sin{\left(x \right)}} x e^{- x^{2}} \log{\left(2 \right)}^{2} \sin^{2}{\left(2^{\sin{\left(x \right)}} \right)} \cos^{2}{\left(x \right)} + 12 \cdot 2^{2 \sin{\left(x \right)}} x e^{- x^{2}} \log{\left(2 \right)}^{2} \cos^{2}{\left(2^{\sin{\left(x \right)}} \right)} \cos^{2}{\left(x \right)} - 6 \cdot 2^{2 \sin{\left(x \right)}} e^{- x^{2}} \log{\left(2 \right)}^{2} \sin^{2}{\left(2^{\sin{\left(x \right)}} \right)} \sin{\left(x \right)} \cos{\left(x \right)} + 6 \cdot 2^{2 \sin{\left(x \right)}} e^{- x^{2}} \log{\left(2 \right)}^{3} \sin^{2}{\left(2^{\sin{\left(x \right)}} \right)} \cos^{3}{\left(x \right)} + 6 \cdot 2^{2 \sin{\left(x \right)}} e^{- x^{2}} \log{\left(2 \right)}^{2} \sin{\left(x \right)} \cos^{2}{\left(2^{\sin{\left(x \right)}} \right)} \cos{\left(x \right)} - 6 \cdot 2^{2 \sin{\left(x \right)}} e^{- x^{2}} \log{\left(2 \right)}^{3} \cos^{2}{\left(2^{\sin{\left(x \right)}} \right)} \cos^{3}{\left(x \right)} - 24 \cdot 2^{\sin{\left(x \right)}} x^{2} e^{- x^{2}} \log{\left(2 \right)} \sin{\left(2^{\sin{\left(x \right)}} \right)} \cos{\left(2^{\sin{\left(x \right)}} \right)} \cos{\left(x \right)} - 12 \cdot 2^{\sin{\left(x \right)}} x e^{- x^{2}} \log{\left(2 \right)} \sin{\left(2^{\sin{\left(x \right)}} \right)} \sin{\left(x \right)} \cos{\left(2^{\sin{\left(x \right)}} \right)} + 12 \cdot 2^{\sin{\left(x \right)}} x e^{- x^{2}} \log{\left(2 \right)}^{2} \sin{\left(2^{\sin{\left(x \right)}} \right)} \cos{\left(2^{\sin{\left(x \right)}} \right)} \cos^{2}{\left(x \right)} + 6 \cdot 2^{\sin{\left(x \right)}} e^{- x^{2}} \log{\left(2 \right)}^{2} \sin{\left(2^{\sin{\left(x \right)}} \right)} \sin{\left(x \right)} \cos{\left(2^{\sin{\left(x \right)}} \right)} \cos{\left(x \right)} - 2 \cdot 2^{\sin{\left(x \right)}} e^{- x^{2}} \log{\left(2 \right)}^{3} \sin{\left(2^{\sin{\left(x \right)}} \right)} \cos{\left(2^{\sin{\left(x \right)}} \right)} \cos^{3}{\left(x \right)} + 14 \cdot 2^{\sin{\left(x \right)}} e^{- x^{2}} \log{\left(2 \right)} \sin{\left(2^{\sin{\left(x \right)}} \right)} \cos{\left(2^{\sin{\left(x \right)}} \right)} \cos{\left(x \right)} - 8 x^{3} e^{- x^{2}} \cos^{2}{\left(2^{\sin{\left(x \right)}} \right)} + 12 x e^{- x^{2}} \cos^{2}{\left(2^{\sin{\left(x \right)}} \right)} + x^{\tan{\left(x \right)}} \left(\left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{\tan{\left(x \right)}}{x}\right)^{3} + 3 x^{\tan{\left(x \right)}} \left(\left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(x \right)} + \frac{\tan{\left(x \right)}}{x}\right) \left(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{\tan{\left(x \right)}}{x^{2}}\right) + x^{\tan{\left(x \right)}} \left(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{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 \tan{\left(x \right)}}{x^{3}}\right)$$