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Derivada de $$$\sin{\left(x^{2} \right)}$$$
Calculadoras relacionadas: Calculadora de diferenciación logarítmica, Calculadora de derivación implícita con pasos
Tu entrada
Halla $$$\frac{d}{dx} \left(\sin{\left(x^{2} \right)}\right)$$$.
Solución
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 $$$g{\left(x \right)} = x^{2}$$$.
Aplica 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 $$$\frac{d}{du} \left(\sin{\left(u \right)}\right) = \cos{\left(u \right)}$$$:
$${\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right)\right)} \frac{d}{dx} \left(x^{2}\right) = {\color{red}\left(\cos{\left(u \right)}\right)} \frac{d}{dx} \left(x^{2}\right)$$Volver a la variable original:
$$\cos{\left({\color{red}\left(u\right)} \right)} \frac{d}{dx} \left(x^{2}\right) = \cos{\left({\color{red}\left(x^{2}\right)} \right)} \frac{d}{dx} \left(x^{2}\right)$$Aplica la regla de la 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, $$$\frac{d}{dx} \left(\sin{\left(x^{2} \right)}\right) = 2 x \cos{\left(x^{2} \right)}$$$.
Respuesta
$$$\frac{d}{dx} \left(\sin{\left(x^{2} \right)}\right) = 2 x \cos{\left(x^{2} \right)}$$$A