Segunda derivada de $$$\sin{\left(x^{2} \right)}$$$
Calculadoras relacionadas: Calculadora de derivados, Calculadora de diferenciación logarítmica
Tu aportación
Encuentra $$$\frac{d^{2}}{dx^{2}} \left(\sin{\left(x^{2} \right)}\right)$$$.
Solución
Encuentra la primera derivada $$$\frac{d}{dx} \left(\sin{\left(x^{2} \right)}\right)$$$
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 $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:
$${\color{red}\left(\frac{d}{dx} \left(\sin{\left(x^{2} \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right) \frac{d}{dx} \left(x^{2}\right)\right)}$$La derivada del seno es $$$\frac{d}{du} \left(\sin{\left(u \right)}\right) = \cos{\left(u \right)}$$$:
$${\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right)\right)} \frac{d}{dx} \left(x^{2}\right) = {\color{red}\left(\cos{\left(u \right)}\right)} \frac{d}{dx} \left(x^{2}\right)$$Vuelva a la variable anterior:
$$\cos{\left({\color{red}\left(u\right)} \right)} \frac{d}{dx} \left(x^{2}\right) = \cos{\left({\color{red}\left(x^{2}\right)} \right)} \frac{d}{dx} \left(x^{2}\right)$$Aplique la regla de potencia $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ con $$$n = 2$$$:
$$\cos{\left(x^{2} \right)} {\color{red}\left(\frac{d}{dx} \left(x^{2}\right)\right)} = \cos{\left(x^{2} \right)} {\color{red}\left(2 x\right)}$$Por lo tanto, $$$\frac{d}{dx} \left(\sin{\left(x^{2} \right)}\right) = 2 x \cos{\left(x^{2} \right)}$$$.
A continuación, $$$\frac{d^{2}}{dx^{2}} \left(\sin{\left(x^{2} \right)}\right) = \frac{d}{dx} \left(2 x \cos{\left(x^{2} \right)}\right)$$$
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 $$$f{\left(x \right)} = x \cos{\left(x^{2} \right)}$$$:
$${\color{red}\left(\frac{d}{dx} \left(2 x \cos{\left(x^{2} \right)}\right)\right)} = {\color{red}\left(2 \frac{d}{dx} \left(x \cos{\left(x^{2} \right)}\right)\right)}$$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 $$$g{\left(x \right)} = \cos{\left(x^{2} \right)}$$$:
$$2 {\color{red}\left(\frac{d}{dx} \left(x \cos{\left(x^{2} \right)}\right)\right)} = 2 {\color{red}\left(\frac{d}{dx} \left(x\right) \cos{\left(x^{2} \right)} + x \frac{d}{dx} \left(\cos{\left(x^{2} \right)}\right)\right)}$$Aplique la regla de potencia $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ con $$$n = 1$$$, en otras palabras, $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$2 x \frac{d}{dx} \left(\cos{\left(x^{2} \right)}\right) + 2 \cos{\left(x^{2} \right)} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = 2 x \frac{d}{dx} \left(\cos{\left(x^{2} \right)}\right) + 2 \cos{\left(x^{2} \right)} {\color{red}\left(1\right)}$$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 $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:
$$2 x {\color{red}\left(\frac{d}{dx} \left(\cos{\left(x^{2} \right)}\right)\right)} + 2 \cos{\left(x^{2} \right)} = 2 x {\color{red}\left(\frac{d}{du} \left(\cos{\left(u \right)}\right) \frac{d}{dx} \left(x^{2}\right)\right)} + 2 \cos{\left(x^{2} \right)}$$La derivada del coseno es $$$\frac{d}{du} \left(\cos{\left(u \right)}\right) = - \sin{\left(u \right)}$$$:
$$2 x {\color{red}\left(\frac{d}{du} \left(\cos{\left(u \right)}\right)\right)} \frac{d}{dx} \left(x^{2}\right) + 2 \cos{\left(x^{2} \right)} = 2 x {\color{red}\left(- \sin{\left(u \right)}\right)} \frac{d}{dx} \left(x^{2}\right) + 2 \cos{\left(x^{2} \right)}$$Vuelva a la variable anterior:
$$- 2 x \sin{\left({\color{red}\left(u\right)} \right)} \frac{d}{dx} \left(x^{2}\right) + 2 \cos{\left(x^{2} \right)} = - 2 x \sin{\left({\color{red}\left(x^{2}\right)} \right)} \frac{d}{dx} \left(x^{2}\right) + 2 \cos{\left(x^{2} \right)}$$Aplique la regla de potencia $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ con $$$n = 2$$$:
$$- 2 x \sin{\left(x^{2} \right)} {\color{red}\left(\frac{d}{dx} \left(x^{2}\right)\right)} + 2 \cos{\left(x^{2} \right)} = - 2 x \sin{\left(x^{2} \right)} {\color{red}\left(2 x\right)} + 2 \cos{\left(x^{2} \right)}$$Por lo tanto, $$$\frac{d}{dx} \left(2 x \cos{\left(x^{2} \right)}\right) = - 4 x^{2} \sin{\left(x^{2} \right)} + 2 \cos{\left(x^{2} \right)}$$$.
Por lo tanto, $$$\frac{d^{2}}{dx^{2}} \left(\sin{\left(x^{2} \right)}\right) = - 4 x^{2} \sin{\left(x^{2} \right)} + 2 \cos{\left(x^{2} \right)}$$$.
Respuesta
$$$\frac{d^{2}}{dx^{2}} \left(\sin{\left(x^{2} \right)}\right) = - 4 x^{2} \sin{\left(x^{2} \right)} + 2 \cos{\left(x^{2} \right)}$$$A