Solución detallada
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Se aplica la regla de la derivada de una multiplicación:
; calculamos :
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Según el principio, aplicamos: tenemos
; calculamos :
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Sustituimos .
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La derivada del seno es igual al coseno:
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Luego se aplica una cadena de reglas. Multiplicamos por :
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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 :
-
La derivada del coseno es igual a menos el seno:
Como resultado de la secuencia de reglas:
Como resultado de la secuencia de reglas:
Como resultado de:
Simplificamos:
Respuesta:
I / I \
I*x*cos (x)*cos\cos (x)/*sin(x) / I \
- ------------------------------- + sin\cos (x)/
cos(x)
$$- \frac{i x \sin{\left(x \right)} \cos^{i}{\left(x \right)} \cos{\left(\cos^{i}{\left(x \right)} \right)}}{\cos{\left(x \right)}} + \sin{\left(\cos^{i}{\left(x \right)} \right)}$$
/ / 2 / I \ 2 / I \ I 2 / I \\ / I \ \
I | | / I \ sin (x)*cos\cos (x)/ I*sin (x)*cos\cos (x)/ cos (x)*sin (x)*sin\cos (x)/| 2*I*cos\cos (x)/*sin(x)|
-cos (x)*|x*|I*cos\cos (x)/ + -------------------- + ---------------------- - ----------------------------| + -----------------------|
| | 2 2 2 | cos(x) |
\ \ cos (x) cos (x) cos (x) / /
$$- \left(x \left(- \frac{\sin^{2}{\left(x \right)} \sin{\left(\cos^{i}{\left(x \right)} \right)} \cos^{i}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{\sin^{2}{\left(x \right)} \cos{\left(\cos^{i}{\left(x \right)} \right)}}{\cos^{2}{\left(x \right)}} + \frac{i \sin^{2}{\left(x \right)} \cos{\left(\cos^{i}{\left(x \right)} \right)}}{\cos^{2}{\left(x \right)}} + i \cos{\left(\cos^{i}{\left(x \right)} \right)}\right) + \frac{2 i \sin{\left(x \right)} \cos{\left(\cos^{i}{\left(x \right)} \right)}}{\cos{\left(x \right)}}\right) \cos^{i}{\left(x \right)}$$
/ / 2 / I \ 2 / I \ I 2 / I \ 2*I 2 / I \ I 2 / I \\ \
| | / I \ I / I \ / I \ 3*sin (x)*cos\cos (x)/ I*sin (x)*cos\cos (x)/ 3*cos (x)*sin (x)*sin\cos (x)/ I*cos (x)*sin (x)*cos\cos (x)/ 3*I*cos (x)*sin (x)*sin\cos (x)/| |
| x*|3*cos\cos (x)/ - 3*cos (x)*sin\cos (x)/ + 2*I*cos\cos (x)/ + ---------------------- + ---------------------- - ------------------------------ + -------------------------------- + --------------------------------|*sin(x) |
| 2 / I \ | 2 2 2 2 2 | I 2 / I \ 2 / I \|
I | / I \ 3*sin (x)*cos\cos (x)/ \ cos (x) cos (x) cos (x) cos (x) cos (x) / 3*cos (x)*sin (x)*sin\cos (x)/ 3*I*sin (x)*cos\cos (x)/|
-cos (x)*|3*I*cos\cos (x)/ + ---------------------- + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ - ------------------------------ + ------------------------|
| 2 cos(x) 2 2 |
\ cos (x) cos (x) cos (x) /
$$- \left(\frac{x \left(- \frac{3 \sin^{2}{\left(x \right)} \sin{\left(\cos^{i}{\left(x \right)} \right)} \cos^{i}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{3 i \sin^{2}{\left(x \right)} \sin{\left(\cos^{i}{\left(x \right)} \right)} \cos^{i}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{i \sin^{2}{\left(x \right)} \cos^{2 i}{\left(x \right)} \cos{\left(\cos^{i}{\left(x \right)} \right)}}{\cos^{2}{\left(x \right)}} + \frac{3 \sin^{2}{\left(x \right)} \cos{\left(\cos^{i}{\left(x \right)} \right)}}{\cos^{2}{\left(x \right)}} + \frac{i \sin^{2}{\left(x \right)} \cos{\left(\cos^{i}{\left(x \right)} \right)}}{\cos^{2}{\left(x \right)}} - 3 \sin{\left(\cos^{i}{\left(x \right)} \right)} \cos^{i}{\left(x \right)} + 3 \cos{\left(\cos^{i}{\left(x \right)} \right)} + 2 i \cos{\left(\cos^{i}{\left(x \right)} \right)}\right) \sin{\left(x \right)}}{\cos{\left(x \right)}} - \frac{3 \sin^{2}{\left(x \right)} \sin{\left(\cos^{i}{\left(x \right)} \right)} \cos^{i}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{3 \sin^{2}{\left(x \right)} \cos{\left(\cos^{i}{\left(x \right)} \right)}}{\cos^{2}{\left(x \right)}} + \frac{3 i \sin^{2}{\left(x \right)} \cos{\left(\cos^{i}{\left(x \right)} \right)}}{\cos^{2}{\left(x \right)}} + 3 i \cos{\left(\cos^{i}{\left(x \right)} \right)}\right) \cos^{i}{\left(x \right)}$$