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