|
|
|
|
|
|
|
|
|
|
|
|
Derivada de $$$e^{\sin{\left(x \right)}}$$$
Calculadoras relacionadas: Calculadora de diferenciación logarítmica, Calculadora de derivación implícita con pasos
Tu entrada
Halla $$$\frac{d}{dx} \left(e^{\sin{\left(x \right)}}\right)$$$.
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 $$$g{\left(x \right)} = \sin{\left(x \right)}$$$.
Aplica la regla de la cadena $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:
$${\color{red}\left(\frac{d}{dx} \left(e^{\sin{\left(x \right)}}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(e^{u}\right) \frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)}$$La derivada de la función 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 variable original:
$$e^{{\color{red}\left(u\right)}} \frac{d}{dx} \left(\sin{\left(x \right)}\right) = e^{{\color{red}\left(\sin{\left(x \right)}\right)}} \frac{d}{dx} \left(\sin{\left(x \right)}\right)$$La derivada del seno es $$$\frac{d}{dx} \left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}$$$:
$$e^{\sin{\left(x \right)}} {\color{red}\left(\frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)} = e^{\sin{\left(x \right)}} {\color{red}\left(\cos{\left(x \right)}\right)}$$Por lo tanto, $$$\frac{d}{dx} \left(e^{\sin{\left(x \right)}}\right) = e^{\sin{\left(x \right)}} \cos{\left(x \right)}$$$.
Respuesta
$$$\frac{d}{dx} \left(e^{\sin{\left(x \right)}}\right) = e^{\sin{\left(x \right)}} \cos{\left(x \right)}$$$A