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Derivada de $$$\sin^{2}{\left(x \right)}$$$
Calculadoras relacionadas: Calculadora de diferenciación logarítmica, Calculadora de derivación implícita con pasos
Tu entrada
Halla $$$\frac{d}{dx} \left(\sin^{2}{\left(x \right)}\right)$$$.
Solución
La función $$$\sin^{2}{\left(x \right)}$$$ es la composición $$$f{\left(g{\left(x \right)} \right)}$$$ de dos funciones $$$f{\left(u \right)} = u^{2}$$$ y $$$g{\left(x \right)} = \sin{\left(x \right)}$$$.
Aplica la regla de la cadena $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:
$${\color{red}\left(\frac{d}{dx} \left(\sin^{2}{\left(x \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(u^{2}\right) \frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)}$$Aplica la regla de la potencia $$$\frac{d}{du} \left(u^{n}\right) = n u^{n - 1}$$$ con $$$n = 2$$$:
$${\color{red}\left(\frac{d}{du} \left(u^{2}\right)\right)} \frac{d}{dx} \left(\sin{\left(x \right)}\right) = {\color{red}\left(2 u\right)} \frac{d}{dx} \left(\sin{\left(x \right)}\right)$$Volver a la variable original:
$$2 {\color{red}\left(u\right)} \frac{d}{dx} \left(\sin{\left(x \right)}\right) = 2 {\color{red}\left(\sin{\left(x \right)}\right)} \frac{d}{dx} \left(\sin{\left(x \right)}\right)$$La derivada del seno es $$$\frac{d}{dx} \left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}$$$:
$$2 \sin{\left(x \right)} {\color{red}\left(\frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)} = 2 \sin{\left(x \right)} {\color{red}\left(\cos{\left(x \right)}\right)}$$Simplificar:
$$2 \sin{\left(x \right)} \cos{\left(x \right)} = \sin{\left(2 x \right)}$$Por lo tanto, $$$\frac{d}{dx} \left(\sin^{2}{\left(x \right)}\right) = \sin{\left(2 x \right)}$$$.
Respuesta
$$$\frac{d}{dx} \left(\sin^{2}{\left(x \right)}\right) = \sin{\left(2 x \right)}$$$A