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Derivada de $$$\cos{\left(5 \theta \right)}$$$
Calculadoras relacionadas: Calculadora de diferenciación logarítmica, Calculadora de derivación implícita con pasos
Tu entrada
Halla $$$\frac{d}{d\theta} \left(\cos{\left(5 \theta \right)}\right)$$$.
Solución
La función $$$\cos{\left(5 \theta \right)}$$$ es la composición $$$f{\left(g{\left(\theta \right)} \right)}$$$ de dos funciones $$$f{\left(u \right)} = \cos{\left(u \right)}$$$ y $$$g{\left(\theta \right)} = 5 \theta$$$.
Aplica la regla de la cadena $$$\frac{d}{d\theta} \left(f{\left(g{\left(\theta \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{d\theta} \left(g{\left(\theta \right)}\right)$$$:
$${\color{red}\left(\frac{d}{d\theta} \left(\cos{\left(5 \theta \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\cos{\left(u \right)}\right) \frac{d}{d\theta} \left(5 \theta\right)\right)}$$La derivada del coseno es $$$\frac{d}{du} \left(\cos{\left(u \right)}\right) = - \sin{\left(u \right)}$$$:
$${\color{red}\left(\frac{d}{du} \left(\cos{\left(u \right)}\right)\right)} \frac{d}{d\theta} \left(5 \theta\right) = {\color{red}\left(- \sin{\left(u \right)}\right)} \frac{d}{d\theta} \left(5 \theta\right)$$Volver a la variable original:
$$- \sin{\left({\color{red}\left(u\right)} \right)} \frac{d}{d\theta} \left(5 \theta\right) = - \sin{\left({\color{red}\left(5 \theta\right)} \right)} \frac{d}{d\theta} \left(5 \theta\right)$$Aplica la regla del factor constante $$$\frac{d}{d\theta} \left(c f{\left(\theta \right)}\right) = c \frac{d}{d\theta} \left(f{\left(\theta \right)}\right)$$$ con $$$c = 5$$$ y $$$f{\left(\theta \right)} = \theta$$$:
$$- \sin{\left(5 \theta \right)} {\color{red}\left(\frac{d}{d\theta} \left(5 \theta\right)\right)} = - \sin{\left(5 \theta \right)} {\color{red}\left(5 \frac{d}{d\theta} \left(\theta\right)\right)}$$Aplica la regla de la potencia $$$\frac{d}{d\theta} \left(\theta^{n}\right) = n \theta^{n - 1}$$$ con $$$n = 1$$$, en otras palabras, $$$\frac{d}{d\theta} \left(\theta\right) = 1$$$:
$$- 5 \sin{\left(5 \theta \right)} {\color{red}\left(\frac{d}{d\theta} \left(\theta\right)\right)} = - 5 \sin{\left(5 \theta \right)} {\color{red}\left(1\right)}$$Por lo tanto, $$$\frac{d}{d\theta} \left(\cos{\left(5 \theta \right)}\right) = - 5 \sin{\left(5 \theta \right)}$$$.
Respuesta
$$$\frac{d}{d\theta} \left(\cos{\left(5 \theta \right)}\right) = - 5 \sin{\left(5 \theta \right)}$$$A