Derivado de $$$\sin^{3}{\left(x \right)}$$$
Calculadoras relacionadas: Calculadora de diferenciación logarítmica, Calculadora de diferenciación implícita con pasos
Tu aportación
Encuentra $$$\frac{d}{dx} \left(\sin^{3}{\left(x \right)}\right)$$$.
Solución
La función $$$\sin^{3}{\left(x \right)}$$$ es la composición $$$f{\left(g{\left(x \right)} \right)}$$$ de dos funciones $$$f{\left(u \right)} = u^{3}$$$ y $$$g{\left(x \right)} = \sin{\left(x \right)}$$$.
Aplicar la regla de la cadena $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:
$${\color{red}\left(\frac{d}{dx} \left(\sin^{3}{\left(x \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(u^{3}\right) \frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)}$$Aplique la regla de potencia $$$\frac{d}{du} \left(u^{n}\right) = n u^{n - 1}$$$ con $$$n = 3$$$:
$${\color{red}\left(\frac{d}{du} \left(u^{3}\right)\right)} \frac{d}{dx} \left(\sin{\left(x \right)}\right) = {\color{red}\left(3 u^{2}\right)} \frac{d}{dx} \left(\sin{\left(x \right)}\right)$$Vuelva a la variable anterior:
$$3 {\color{red}\left(u\right)}^{2} \frac{d}{dx} \left(\sin{\left(x \right)}\right) = 3 {\color{red}\left(\sin{\left(x \right)}\right)}^{2} \frac{d}{dx} \left(\sin{\left(x \right)}\right)$$La derivada del seno es $$$\frac{d}{dx} \left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}$$$:
$$3 \sin^{2}{\left(x \right)} {\color{red}\left(\frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)} = 3 \sin^{2}{\left(x \right)} {\color{red}\left(\cos{\left(x \right)}\right)}$$Por lo tanto, $$$\frac{d}{dx} \left(\sin^{3}{\left(x \right)}\right) = 3 \sin^{2}{\left(x \right)} \cos{\left(x \right)}$$$.
Respuesta
$$$\frac{d}{dx} \left(\sin^{3}{\left(x \right)}\right) = 3 \sin^{2}{\left(x \right)} \cos{\left(x \right)}$$$A