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Segunda derivada de $$$\cos{\left(x^{2} \right)}$$$

La calculadora encontrará la segunda derivada de $$$\cos{\left(x^{2} \right)}$$$, con los pasos que se muestran.

Calculadoras relacionadas: Calculadora de derivados, Calculadora de diferenciación logarítmica

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Tu aportación

Encuentra $$$\frac{d^{2}}{dx^{2}} \left(\cos{\left(x^{2} \right)}\right)$$$.

Solución

Encuentra la primera derivada $$$\frac{d}{dx} \left(\cos{\left(x^{2} \right)}\right)$$$

La función $$$\cos{\left(x^{2} \right)}$$$ es la composición $$$f{\left(g{\left(x \right)} \right)}$$$ de dos funciones $$$f{\left(u \right)} = \cos{\left(u \right)}$$$ y $$$g{\left(x \right)} = x^{2}$$$.

Aplicar la regla de la cadena $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:

$${\color{red}\left(\frac{d}{dx} \left(\cos{\left(x^{2} \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\cos{\left(u \right)}\right) \frac{d}{dx} \left(x^{2}\right)\right)}$$

La derivada del coseno es $$$\frac{d}{du} \left(\cos{\left(u \right)}\right) = - \sin{\left(u \right)}$$$:

$${\color{red}\left(\frac{d}{du} \left(\cos{\left(u \right)}\right)\right)} \frac{d}{dx} \left(x^{2}\right) = {\color{red}\left(- \sin{\left(u \right)}\right)} \frac{d}{dx} \left(x^{2}\right)$$

Vuelva a la variable anterior:

$$- \sin{\left({\color{red}\left(u\right)} \right)} \frac{d}{dx} \left(x^{2}\right) = - \sin{\left({\color{red}\left(x^{2}\right)} \right)} \frac{d}{dx} \left(x^{2}\right)$$

Aplique la regla de potencia $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ con $$$n = 2$$$:

$$- \sin{\left(x^{2} \right)} {\color{red}\left(\frac{d}{dx} \left(x^{2}\right)\right)} = - \sin{\left(x^{2} \right)} {\color{red}\left(2 x\right)}$$

Por lo tanto, $$$\frac{d}{dx} \left(\cos{\left(x^{2} \right)}\right) = - 2 x \sin{\left(x^{2} \right)}$$$.

A continuación, $$$\frac{d^{2}}{dx^{2}} \left(\cos{\left(x^{2} \right)}\right) = \frac{d}{dx} \left(- 2 x \sin{\left(x^{2} \right)}\right)$$$

Aplique la regla del múltiplo constante $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ con $$$c = -2$$$ y $$$f{\left(x \right)} = x \sin{\left(x^{2} \right)}$$$:

$${\color{red}\left(\frac{d}{dx} \left(- 2 x \sin{\left(x^{2} \right)}\right)\right)} = {\color{red}\left(- 2 \frac{d}{dx} \left(x \sin{\left(x^{2} \right)}\right)\right)}$$

Aplique la regla del producto $$$\frac{d}{dx} \left(f{\left(x \right)} g{\left(x \right)}\right) = \frac{d}{dx} \left(f{\left(x \right)}\right) g{\left(x \right)} + f{\left(x \right)} \frac{d}{dx} \left(g{\left(x \right)}\right)$$$ con $$$f{\left(x \right)} = x$$$ y $$$g{\left(x \right)} = \sin{\left(x^{2} \right)}$$$:

$$- 2 {\color{red}\left(\frac{d}{dx} \left(x \sin{\left(x^{2} \right)}\right)\right)} = - 2 {\color{red}\left(\frac{d}{dx} \left(x\right) \sin{\left(x^{2} \right)} + x \frac{d}{dx} \left(\sin{\left(x^{2} \right)}\right)\right)}$$

La función $$$\sin{\left(x^{2} \right)}$$$ es la composición $$$f{\left(g{\left(x \right)} \right)}$$$ de dos funciones $$$f{\left(u \right)} = \sin{\left(u \right)}$$$ y $$$g{\left(x \right)} = x^{2}$$$.

Aplicar la regla de la cadena $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:

$$- 2 x {\color{red}\left(\frac{d}{dx} \left(\sin{\left(x^{2} \right)}\right)\right)} - 2 \sin{\left(x^{2} \right)} \frac{d}{dx} \left(x\right) = - 2 x {\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right) \frac{d}{dx} \left(x^{2}\right)\right)} - 2 \sin{\left(x^{2} \right)} \frac{d}{dx} \left(x\right)$$

La derivada del seno es $$$\frac{d}{du} \left(\sin{\left(u \right)}\right) = \cos{\left(u \right)}$$$:

$$- 2 x {\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right)\right)} \frac{d}{dx} \left(x^{2}\right) - 2 \sin{\left(x^{2} \right)} \frac{d}{dx} \left(x\right) = - 2 x {\color{red}\left(\cos{\left(u \right)}\right)} \frac{d}{dx} \left(x^{2}\right) - 2 \sin{\left(x^{2} \right)} \frac{d}{dx} \left(x\right)$$

Vuelva a la variable anterior:

$$- 2 x \cos{\left({\color{red}\left(u\right)} \right)} \frac{d}{dx} \left(x^{2}\right) - 2 \sin{\left(x^{2} \right)} \frac{d}{dx} \left(x\right) = - 2 x \cos{\left({\color{red}\left(x^{2}\right)} \right)} \frac{d}{dx} \left(x^{2}\right) - 2 \sin{\left(x^{2} \right)} \frac{d}{dx} \left(x\right)$$

Aplique la regla de potencia $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ con $$$n = 2$$$:

$$- 2 x \cos{\left(x^{2} \right)} {\color{red}\left(\frac{d}{dx} \left(x^{2}\right)\right)} - 2 \sin{\left(x^{2} \right)} \frac{d}{dx} \left(x\right) = - 2 x \cos{\left(x^{2} \right)} {\color{red}\left(2 x\right)} - 2 \sin{\left(x^{2} \right)} \frac{d}{dx} \left(x\right)$$

Aplique la regla de potencia $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ con $$$n = 1$$$, en otras palabras, $$$\frac{d}{dx} \left(x\right) = 1$$$:

$$- 4 x^{2} \cos{\left(x^{2} \right)} - 2 \sin{\left(x^{2} \right)} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = - 4 x^{2} \cos{\left(x^{2} \right)} - 2 \sin{\left(x^{2} \right)} {\color{red}\left(1\right)}$$

Por lo tanto, $$$\frac{d}{dx} \left(- 2 x \sin{\left(x^{2} \right)}\right) = - 4 x^{2} \cos{\left(x^{2} \right)} - 2 \sin{\left(x^{2} \right)}$$$.

Por lo tanto, $$$\frac{d^{2}}{dx^{2}} \left(\cos{\left(x^{2} \right)}\right) = - 4 x^{2} \cos{\left(x^{2} \right)} - 2 \sin{\left(x^{2} \right)}$$$.

Respuesta

$$$\frac{d^{2}}{dx^{2}} \left(\cos{\left(x^{2} \right)}\right) = - 4 x^{2} \cos{\left(x^{2} \right)} - 2 \sin{\left(x^{2} \right)}$$$A