Solución detallada
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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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Se aplica la regla de la derivada parcial:
y .
Para calcular :
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Se aplica la regla de la derivada de una multiplicación:
; calculamos :
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; calculamos :
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La derivada del coseno es igual a menos el seno:
; calculamos :
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La derivada del seno es igual al coseno:
Como resultado de:
Para calcular :
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Derivado es.
Ahora aplicamos la regla de la derivada de una divesión:
Como resultado de la secuencia de reglas:
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Simplificamos:
Respuesta:
-x / x // 2 2 \ -x -x \ x -x \ x
2 *\2 *\\cos (x) - sin (x)/*e - cos(x)*e *sin(x)/ + 2 *cos(x)*e *log(2)*sin(x)/*e
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cos(x)*sin(x)
$$\frac{2^{- x} \left(2^{x} \left(\left(- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) e^{- x} - e^{- x} \sin{\left(x \right)} \cos{\left(x \right)}\right) + 2^{x} e^{- x} \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)}\right) e^{x}}{\sin{\left(x \right)} \cos{\left(x \right)}}$$
2 2 2 2 2 2 2 2 2 / 2 2 \ 2 2 / 2 2 \
sin (x) - cos (x) + cos(x)*sin(x) - cos(x)*log(2)*sin(x) sin (x) - cos (x) + cos(x)*sin(x) - cos(x)*log(2)*sin(x) sin (x) - cos (x) + cos(x)*sin(x) - cos(x)*log(2)*sin(x) - 2*sin (x) + 2*cos (x) + sin (x)*log(2) + \sin (x) - cos (x) + cos(x)*sin(x)/*log(2) - cos (x)*log(2) + 3*cos(x)*sin(x) + cos(x)*log(2)*sin(x) - log (2)*cos(x)*sin(x) \sin (x) - cos (x) + cos(x)*sin(x) - cos(x)*log(2)*sin(x)/*log(2)
-------------------------------------------------------- - -------------------------------------------------------- - -------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------------------------- + -----------------------------------------------------------------
2 2 cos(x)*sin(x) cos(x)*sin(x) cos(x)*sin(x)
sin (x) cos (x)
$$- \frac{\sin^{2}{\left(x \right)} - \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} - \frac{\sin^{2}{\left(x \right)} - \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} \cos{\left(x \right)}} + \frac{\left(\sin^{2}{\left(x \right)} - \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)}}{\sin{\left(x \right)} \cos{\left(x \right)}} + \frac{\sin^{2}{\left(x \right)} - \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{\left(\sin^{2}{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)} - 2 \sin^{2}{\left(x \right)} + \log{\left(2 \right)} \sin^{2}{\left(x \right)} - \log{\left(2 \right)}^{2} \sin{\left(x \right)} \cos{\left(x \right)} + \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + 3 \sin{\left(x \right)} \cos{\left(x \right)} - \log{\left(2 \right)} \cos^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} \cos{\left(x \right)}}$$
/ 2 2 \ / 2 2 2 / 2 2 \ 2 2 \ / 2 2 \ / 2 2 2 / 2 2 \ 2 2 \ 2 2 2 2 2 / 2 2 \ 2 2 2 / 2 2 \ 2 2 2 3 2 / 2 2 2 / 2 2 \ 2 2 \ / 2 2 \ / 2 2 \ / 2 2 \ / 2 2 \ 2 / 2 2 \ / 2 2 \ / 2 2 2 / 2 2 \ 2 2 \
2*\sin (x) - cos (x) + cos(x)*sin(x) - cos(x)*log(2)*sin(x)/ 2*\- 2*sin (x) + 2*cos (x) + sin (x)*log(2) + \sin (x) - cos (x) + cos(x)*sin(x)/*log(2) - cos (x)*log(2) + 3*cos(x)*sin(x) + cos(x)*log(2)*sin(x) - log (2)*cos(x)*sin(x)/ 2*\sin (x) - cos (x) + cos(x)*sin(x) - cos(x)*log(2)*sin(x)/ 2*\- 2*sin (x) + 2*cos (x) + sin (x)*log(2) + \sin (x) - cos (x) + cos(x)*sin(x)/*log(2) - cos (x)*log(2) + 3*cos(x)*sin(x) + cos(x)*log(2)*sin(x) - log (2)*cos(x)*sin(x)/ sin (x) - cos (x) + cos(x)*sin(x) - cos(x)*log(2)*sin(x) cos (x) - sin (x) + log (2)*\sin (x) - cos (x) + cos(x)*sin(x)/ - 11*cos(x)*sin(x) - 2*cos (x)*log (2) - 2*sin (x)*log(2) - 2*\- 2*cos (x) + 2*sin (x) - 3*cos(x)*sin(x)/*log(2) + 2*cos (x)*log(2) + 2*log (2)*sin (x) - log (2)*cos(x)*sin(x) + 2*log (2)*cos(x)*sin(x) + 3*cos(x)*log(2)*sin(x) 2*\- 2*sin (x) + 2*cos (x) + sin (x)*log(2) + \sin (x) - cos (x) + cos(x)*sin(x)/*log(2) - cos (x)*log(2) + 3*cos(x)*sin(x) + cos(x)*log(2)*sin(x) - log (2)*cos(x)*sin(x)/ 2*\sin (x) - cos (x) + cos(x)*sin(x) - cos(x)*log(2)*sin(x)/*sin(x) 2*\sin (x) - cos (x) + cos(x)*sin(x) - cos(x)*log(2)*sin(x)/*cos(x) 2*\sin (x) - cos (x) + cos(x)*sin(x) - cos(x)*log(2)*sin(x)/*log(2) 2*\sin (x) - cos (x) + cos(x)*sin(x) - cos(x)*log(2)*sin(x)/*log(2) log (2)*\sin (x) - cos (x) + cos(x)*sin(x) - cos(x)*log(2)*sin(x)/ 2*\sin (x) - cos (x) + cos(x)*sin(x) - cos(x)*log(2)*sin(x)/*log(2) 2*\- 2*sin (x) + 2*cos (x) + sin (x)*log(2) + \sin (x) - cos (x) + cos(x)*sin(x)/*log(2) - cos (x)*log(2) + 3*cos(x)*sin(x) + cos(x)*log(2)*sin(x) - log (2)*cos(x)*sin(x)/*log(2)
- ------------------------------------------------------------ - --------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ------------------------------------------------------------ + --------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - -------------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------- - ------------------------------------------------------------------- - ------------------------------------------------------------------- + ------------------------------------------------------------------- - ------------------------------------------------------------------ + ------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
2 2 2 2 cos(x)*sin(x) cos(x)*sin(x) cos(x)*sin(x) 3 3 2 2 cos(x)*sin(x) cos(x)*sin(x) cos(x)*sin(x)
cos (x) cos (x) sin (x) sin (x) cos (x) sin (x) sin (x) cos (x)
$$- \frac{2 \left(\sin^{2}{\left(x \right)} - \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)}}{\cos^{3}{\left(x \right)}} - \frac{2 \left(\sin^{2}{\left(x \right)} - \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + \frac{2 \left(\sin^{2}{\left(x \right)} - \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)}}{\cos^{2}{\left(x \right)}} - \frac{\sin^{2}{\left(x \right)} - \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} \cos{\left(x \right)}} - \frac{\left(\sin^{2}{\left(x \right)} - \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)}^{2}}{\sin{\left(x \right)} \cos{\left(x \right)}} + \frac{2 \left(\sin^{2}{\left(x \right)} - \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)}}{\sin{\left(x \right)} \cos{\left(x \right)}} - \frac{2 \left(\sin^{2}{\left(x \right)} - \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)}}{\sin^{2}{\left(x \right)}} + \frac{2 \left(\sin^{2}{\left(x \right)} - \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right)}{\sin^{2}{\left(x \right)}} - \frac{2 \left(\sin^{2}{\left(x \right)} - \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \cos{\left(x \right)}}{\sin^{3}{\left(x \right)}} - \frac{2 \left(\left(\sin^{2}{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)} - 2 \sin^{2}{\left(x \right)} + \log{\left(2 \right)} \sin^{2}{\left(x \right)} - \log{\left(2 \right)}^{2} \sin{\left(x \right)} \cos{\left(x \right)} + \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + 3 \sin{\left(x \right)} \cos{\left(x \right)} - \log{\left(2 \right)} \cos^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} - \frac{2 \left(\left(\sin^{2}{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)} - 2 \sin^{2}{\left(x \right)} + \log{\left(2 \right)} \sin^{2}{\left(x \right)} - \log{\left(2 \right)}^{2} \sin{\left(x \right)} \cos{\left(x \right)} + \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + 3 \sin{\left(x \right)} \cos{\left(x \right)} - \log{\left(2 \right)} \cos^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}\right)}{\sin{\left(x \right)} \cos{\left(x \right)}} + \frac{2 \left(\left(\sin^{2}{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)} - 2 \sin^{2}{\left(x \right)} + \log{\left(2 \right)} \sin^{2}{\left(x \right)} - \log{\left(2 \right)}^{2} \sin{\left(x \right)} \cos{\left(x \right)} + \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + 3 \sin{\left(x \right)} \cos{\left(x \right)} - \log{\left(2 \right)} \cos^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)}}{\sin{\left(x \right)} \cos{\left(x \right)}} + \frac{2 \left(\left(\sin^{2}{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)} - 2 \sin^{2}{\left(x \right)} + \log{\left(2 \right)} \sin^{2}{\left(x \right)} - \log{\left(2 \right)}^{2} \sin{\left(x \right)} \cos{\left(x \right)} + \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} + 3 \sin{\left(x \right)} \cos{\left(x \right)} - \log{\left(2 \right)} \cos^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}\right)}{\sin^{2}{\left(x \right)}} - \frac{\left(\sin^{2}{\left(x \right)} + \sin{\left(x \right)} \cos{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)}^{2} - 2 \left(2 \sin^{2}{\left(x \right)} - 3 \sin{\left(x \right)} \cos{\left(x \right)} - 2 \cos^{2}{\left(x \right)}\right) \log{\left(2 \right)} - 2 \log{\left(2 \right)} \sin^{2}{\left(x \right)} - \sin^{2}{\left(x \right)} + 2 \log{\left(2 \right)}^{2} \sin^{2}{\left(x \right)} - 11 \sin{\left(x \right)} \cos{\left(x \right)} - \log{\left(2 \right)}^{3} \sin{\left(x \right)} \cos{\left(x \right)} + 2 \log{\left(2 \right)}^{2} \sin{\left(x \right)} \cos{\left(x \right)} + 3 \log{\left(2 \right)} \sin{\left(x \right)} \cos{\left(x \right)} - 2 \log{\left(2 \right)}^{2} \cos^{2}{\left(x \right)} + \cos^{2}{\left(x \right)} + 2 \log{\left(2 \right)} \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} \cos{\left(x \right)}}$$