Derivado de $e^{\sin{\left(x \right)}}$

La calculadora hallará la derivada de

e^{\sin{\left(x \right)}}

, con los pasos mostrados.

Calculadora relacionada: Calculadora de derivadas

Solución

La función $e^{\sin{\left(x \right)}}$ es la composición $f{\left(g{\left(x \right)} \right)}$ de dos funciones $f{\left(u \right)} = e^{u}$ y $g{\left(x \right)} = \sin{\left(x \right)}$ .

Aplique 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(e^{\sin{\left(x \right)}}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(e^{u}\right) \frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)}

La derivada de la exponencial es $\frac{d}{du} \left(e^{u}\right) = e^{u}$ :

{\color{red}\left(\frac{d}{du} \left(e^{u}\right)\right)} \frac{d}{dx} \left(\sin{\left(x \right)}\right) = {\color{red}\left(e^{u}\right)} \frac{d}{dx} \left(\sin{\left(x \right)}\right)

Volver a la antigua variable:

e^{{\color{red}\left(u\right)}} \frac{d}{dx} \left(\sin{\left(x \right)}\right) = e^{{\color{red}\left(\sin{\left(x \right)}\right)}} \frac{d}{dx} \left(\sin{\left(x \right)}\right)

La derivada del seno es $\frac{d}{dx} \left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}$ :

e^{\sin{\left(x \right)}} {\color{red}\left(\frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)} = e^{\sin{\left(x \right)}} {\color{red}\left(\cos{\left(x \right)}\right)}

Así, $\frac{d}{dx} \left(e^{\sin{\left(x \right)}}\right) = e^{\sin{\left(x \right)}} \cos{\left(x \right)}$ .

Respuesta

$\frac{d}{dx} \left(e^{\sin{\left(x \right)}}\right) = e^{\sin{\left(x \right)}} \cos{\left(x \right)}$ A

Derivado de esin⁡(x)e^{\sin{\left(x \right)}}esin(x)

Solución

Respuesta

Derivado de $e^{\sin{\left(x \right)}}$