Derivada de $$$\ln\left(32 y\right)$$$
Calculadoras relacionadas: Calculadora de diferenciación logarítmica, Calculadora de derivación implícita con pasos
Tu entrada
Halla $$$\frac{d}{dy} \left(\ln\left(32 y\right)\right)$$$.
Solución
La función $$$\ln\left(32 y\right)$$$ es la composición $$$f{\left(g{\left(y \right)} \right)}$$$ de dos funciones $$$f{\left(u \right)} = \ln\left(u\right)$$$ y $$$g{\left(y \right)} = 32 y$$$.
Aplica la regla de la cadena $$$\frac{d}{dy} \left(f{\left(g{\left(y \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dy} \left(g{\left(y \right)}\right)$$$:
$${\color{red}\left(\frac{d}{dy} \left(\ln\left(32 y\right)\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right) \frac{d}{dy} \left(32 y\right)\right)}$$La derivada del logaritmo natural es $$$\frac{d}{du} \left(\ln\left(u\right)\right) = \frac{1}{u}$$$:
$${\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right)\right)} \frac{d}{dy} \left(32 y\right) = {\color{red}\left(\frac{1}{u}\right)} \frac{d}{dy} \left(32 y\right)$$Volver a la variable original:
$$\frac{\frac{d}{dy} \left(32 y\right)}{{\color{red}\left(u\right)}} = \frac{\frac{d}{dy} \left(32 y\right)}{{\color{red}\left(32 y\right)}}$$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 $$$c = 32$$$ y $$$f{\left(y \right)} = y$$$:
$$\frac{{\color{red}\left(\frac{d}{dy} \left(32 y\right)\right)}}{32 y} = \frac{{\color{red}\left(32 \frac{d}{dy} \left(y\right)\right)}}{32 y}$$Aplica la regla de la potencia $$$\frac{d}{dy} \left(y^{n}\right) = n y^{n - 1}$$$ con $$$n = 1$$$, en otras palabras, $$$\frac{d}{dy} \left(y\right) = 1$$$:
$$\frac{{\color{red}\left(\frac{d}{dy} \left(y\right)\right)}}{y} = \frac{{\color{red}\left(1\right)}}{y}$$Por lo tanto, $$$\frac{d}{dy} \left(\ln\left(32 y\right)\right) = \frac{1}{y}$$$.
Respuesta
$$$\frac{d}{dy} \left(\ln\left(32 y\right)\right) = \frac{1}{y}$$$A