Derivada de $$$e^{x} \cos{\left(x \right)}$$$
Calculadoras relacionadas: Calculadora de diferenciación logarítmica, Calculadora de derivación implícita con pasos
Tu entrada
Halla $$$\frac{d}{dx} \left(e^{x} \cos{\left(x \right)}\right)$$$.
Solución
Aplica la regla del producto $$$\frac{d}{dx} \left(f{\left(x \right)} g{\left(x \right)}\right) = \frac{d}{dx} \left(f{\left(x \right)}\right) g{\left(x \right)} + f{\left(x \right)} \frac{d}{dx} \left(g{\left(x \right)}\right)$$$ con $$$f{\left(x \right)} = \cos{\left(x \right)}$$$ y $$$g{\left(x \right)} = e^{x}$$$:
$${\color{red}\left(\frac{d}{dx} \left(e^{x} \cos{\left(x \right)}\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(\cos{\left(x \right)}\right) e^{x} + \cos{\left(x \right)} \frac{d}{dx} \left(e^{x}\right)\right)}$$La derivada del coseno es $$$\frac{d}{dx} \left(\cos{\left(x \right)}\right) = - \sin{\left(x \right)}$$$:
$$e^{x} {\color{red}\left(\frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)} + \cos{\left(x \right)} \frac{d}{dx} \left(e^{x}\right) = e^{x} {\color{red}\left(- \sin{\left(x \right)}\right)} + \cos{\left(x \right)} \frac{d}{dx} \left(e^{x}\right)$$La derivada de la función exponencial es $$$\frac{d}{dx} \left(e^{x}\right) = e^{x}$$$:
$$- e^{x} \sin{\left(x \right)} + \cos{\left(x \right)} {\color{red}\left(\frac{d}{dx} \left(e^{x}\right)\right)} = - e^{x} \sin{\left(x \right)} + \cos{\left(x \right)} {\color{red}\left(e^{x}\right)}$$Simplificar:
$$- e^{x} \sin{\left(x \right)} + e^{x} \cos{\left(x \right)} = \sqrt{2} e^{x} \cos{\left(x + \frac{\pi}{4} \right)}$$Por lo tanto, $$$\frac{d}{dx} \left(e^{x} \cos{\left(x \right)}\right) = \sqrt{2} e^{x} \cos{\left(x + \frac{\pi}{4} \right)}$$$.
Respuesta
$$$\frac{d}{dx} \left(e^{x} \cos{\left(x \right)}\right) = \sqrt{2} e^{x} \cos{\left(x + \frac{\pi}{4} \right)}$$$A