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Derivada de $$$e^{x} + \sin{\left(y z \right)}$$$ con respecto a $$$y$$$
Calculadoras relacionadas: Calculadora de diferenciación logarítmica, Calculadora de derivación implícita con pasos
Tu entrada
Halla $$$\frac{d}{dy} \left(e^{x} + \sin{\left(y z \right)}\right)$$$.
Solución
La derivada de una suma/diferencia es la suma/diferencia de las derivadas:
$${\color{red}\left(\frac{d}{dy} \left(e^{x} + \sin{\left(y z \right)}\right)\right)} = {\color{red}\left(\frac{d}{dy} \left(e^{x}\right) + \frac{d}{dy} \left(\sin{\left(y z \right)}\right)\right)}$$La función $$$\sin{\left(y z \right)}$$$ es la composición $$$f{\left(g{\left(y \right)} \right)}$$$ de dos funciones $$$f{\left(u \right)} = \sin{\left(u \right)}$$$ y $$$g{\left(y \right)} = y z$$$.
Aplica la regla de la cadena $$$\frac{d}{dy} \left(f{\left(g{\left(y \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dy} \left(g{\left(y \right)}\right)$$$:
$${\color{red}\left(\frac{d}{dy} \left(\sin{\left(y z \right)}\right)\right)} + \frac{d}{dy} \left(e^{x}\right) = {\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right) \frac{d}{dy} \left(y z\right)\right)} + \frac{d}{dy} \left(e^{x}\right)$$La derivada del seno es $$$\frac{d}{du} \left(\sin{\left(u \right)}\right) = \cos{\left(u \right)}$$$:
$${\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right)\right)} \frac{d}{dy} \left(y z\right) + \frac{d}{dy} \left(e^{x}\right) = {\color{red}\left(\cos{\left(u \right)}\right)} \frac{d}{dy} \left(y z\right) + \frac{d}{dy} \left(e^{x}\right)$$Volver a la variable original:
$$\cos{\left({\color{red}\left(u\right)} \right)} \frac{d}{dy} \left(y z\right) + \frac{d}{dy} \left(e^{x}\right) = \cos{\left({\color{red}\left(y z\right)} \right)} \frac{d}{dy} \left(y z\right) + \frac{d}{dy} \left(e^{x}\right)$$Aplica la regla del factor constante $$$\frac{d}{dy} \left(c f{\left(y \right)}\right) = c \frac{d}{dy} \left(f{\left(y \right)}\right)$$$ con $$$c = z$$$ y $$$f{\left(y \right)} = y$$$:
$$\cos{\left(y z \right)} {\color{red}\left(\frac{d}{dy} \left(y z\right)\right)} + \frac{d}{dy} \left(e^{x}\right) = \cos{\left(y z \right)} {\color{red}\left(z \frac{d}{dy} \left(y\right)\right)} + \frac{d}{dy} \left(e^{x}\right)$$La derivada de una constante es $$$0$$$:
$$z \cos{\left(y z \right)} \frac{d}{dy} \left(y\right) + {\color{red}\left(\frac{d}{dy} \left(e^{x}\right)\right)} = z \cos{\left(y z \right)} \frac{d}{dy} \left(y\right) + {\color{red}\left(0\right)}$$Aplica la regla de la potencia $$$\frac{d}{dy} \left(y^{n}\right) = n y^{n - 1}$$$ con $$$n = 1$$$, en otras palabras, $$$\frac{d}{dy} \left(y\right) = 1$$$:
$$z \cos{\left(y z \right)} {\color{red}\left(\frac{d}{dy} \left(y\right)\right)} = z \cos{\left(y z \right)} {\color{red}\left(1\right)}$$Por lo tanto, $$$\frac{d}{dy} \left(e^{x} + \sin{\left(y z \right)}\right) = z \cos{\left(y z \right)}$$$.
Respuesta
$$$\frac{d}{dy} \left(e^{x} + \sin{\left(y z \right)}\right) = z \cos{\left(y z \right)}$$$A