Derivado de $$$\left(x^{3} + 4\right)^{4} \left(x^{5} + 2\right)^{2}$$$

Sea $$$H{\left(x \right)} = \left(x^{3} + 4\right)^{4} \left(x^{5} + 2\right)^{2}$$$.

Toma el logaritmo de ambos lados: $$$\ln\left(H{\left(x \right)}\right) = \ln\left(\left(x^{3} + 4\right)^{4} \left(x^{5} + 2\right)^{2}\right)$$$.

Vuelve a escribir la RHS usando las propiedades de los logaritmos: $$$\ln\left(H{\left(x \right)}\right) = 4 \ln\left(x^{3} + 4\right) + 2 \ln\left(x^{5} + 2\right)$$$.

Derive por separado ambos lados de la ecuación: $$$\frac{d}{dx} \left(\ln\left(H{\left(x \right)}\right)\right) = \frac{d}{dx} \left(4 \ln\left(x^{3} + 4\right) + 2 \ln\left(x^{5} + 2\right)\right)$$$.

Diferenciar el LHS de la ecuación.

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

Aplicar la regla de la cadena $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:

$${\color{red}\left(\frac{d}{dx} \left(\ln\left(H{\left(x \right)}\right)\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right) \frac{d}{dx} \left(H{\left(x \right)}\right)\right)}$$

La derivada del logaritmo natural es $$$\frac{d}{du} \left(\ln\left(u\right)\right) = \frac{1}{u}$$$:

$${\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right)\right)} \frac{d}{dx} \left(H{\left(x \right)}\right) = {\color{red}\left(\frac{1}{u}\right)} \frac{d}{dx} \left(H{\left(x \right)}\right)$$

Vuelva a la variable anterior:

$$\frac{\frac{d}{dx} \left(H{\left(x \right)}\right)}{{\color{red}\left(u\right)}} = \frac{\frac{d}{dx} \left(H{\left(x \right)}\right)}{{\color{red}\left(H{\left(x \right)}\right)}}$$

Por lo tanto, $$$\frac{d}{dx} \left(\ln\left(H{\left(x \right)}\right)\right) = \frac{\frac{d}{dx} \left(H{\left(x \right)}\right)}{H{\left(x \right)}}$$$.

Derive la RHS de la ecuación.

La derivada de una suma/diferencia es la suma/diferencia de derivadas:

$${\color{red}\left(\frac{d}{dx} \left(4 \ln\left(x^{3} + 4\right) + 2 \ln\left(x^{5} + 2\right)\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(4 \ln\left(x^{3} + 4\right)\right) + \frac{d}{dx} \left(2 \ln\left(x^{5} + 2\right)\right)\right)}$$

Aplique la regla del múltiplo constante $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ con $$$c = 2$$$ y $$$f{\left(x \right)} = \ln\left(x^{5} + 2\right)$$$:

$${\color{red}\left(\frac{d}{dx} \left(2 \ln\left(x^{5} + 2\right)\right)\right)} + \frac{d}{dx} \left(4 \ln\left(x^{3} + 4\right)\right) = {\color{red}\left(2 \frac{d}{dx} \left(\ln\left(x^{5} + 2\right)\right)\right)} + \frac{d}{dx} \left(4 \ln\left(x^{3} + 4\right)\right)$$

La función $$$\ln\left(x^{5} + 2\right)$$$ es la composición $$$f{\left(g{\left(x \right)} \right)}$$$ de dos funciones $$$f{\left(u \right)} = \ln\left(u\right)$$$ y $$$g{\left(x \right)} = x^{5} + 2$$$.

Aplicar la regla de la cadena $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:

$$2 {\color{red}\left(\frac{d}{dx} \left(\ln\left(x^{5} + 2\right)\right)\right)} + \frac{d}{dx} \left(4 \ln\left(x^{3} + 4\right)\right) = 2 {\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right) \frac{d}{dx} \left(x^{5} + 2\right)\right)} + \frac{d}{dx} \left(4 \ln\left(x^{3} + 4\right)\right)$$

La derivada del logaritmo natural es $$$\frac{d}{du} \left(\ln\left(u\right)\right) = \frac{1}{u}$$$:

$$2 {\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right)\right)} \frac{d}{dx} \left(x^{5} + 2\right) + \frac{d}{dx} \left(4 \ln\left(x^{3} + 4\right)\right) = 2 {\color{red}\left(\frac{1}{u}\right)} \frac{d}{dx} \left(x^{5} + 2\right) + \frac{d}{dx} \left(4 \ln\left(x^{3} + 4\right)\right)$$

Vuelva a la variable anterior:

$$\frac{d}{dx} \left(4 \ln\left(x^{3} + 4\right)\right) + \frac{2 \frac{d}{dx} \left(x^{5} + 2\right)}{{\color{red}\left(u\right)}} = \frac{d}{dx} \left(4 \ln\left(x^{3} + 4\right)\right) + \frac{2 \frac{d}{dx} \left(x^{5} + 2\right)}{{\color{red}\left(x^{5} + 2\right)}}$$

Aplique la regla del múltiplo constante $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ con $$$c = 4$$$ y $$$f{\left(x \right)} = \ln\left(x^{3} + 4\right)$$$:

$${\color{red}\left(\frac{d}{dx} \left(4 \ln\left(x^{3} + 4\right)\right)\right)} + \frac{2 \frac{d}{dx} \left(x^{5} + 2\right)}{x^{5} + 2} = {\color{red}\left(4 \frac{d}{dx} \left(\ln\left(x^{3} + 4\right)\right)\right)} + \frac{2 \frac{d}{dx} \left(x^{5} + 2\right)}{x^{5} + 2}$$

La función $$$\ln\left(x^{3} + 4\right)$$$ es la composición $$$f{\left(g{\left(x \right)} \right)}$$$ de dos funciones $$$f{\left(u \right)} = \ln\left(u\right)$$$ y $$$g{\left(x \right)} = x^{3} + 4$$$.

Aplicar la regla de la cadena $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:

$$4 {\color{red}\left(\frac{d}{dx} \left(\ln\left(x^{3} + 4\right)\right)\right)} + \frac{2 \frac{d}{dx} \left(x^{5} + 2\right)}{x^{5} + 2} = 4 {\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right) \frac{d}{dx} \left(x^{3} + 4\right)\right)} + \frac{2 \frac{d}{dx} \left(x^{5} + 2\right)}{x^{5} + 2}$$

La derivada del logaritmo natural es $$$\frac{d}{du} \left(\ln\left(u\right)\right) = \frac{1}{u}$$$:

$$4 {\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right)\right)} \frac{d}{dx} \left(x^{3} + 4\right) + \frac{2 \frac{d}{dx} \left(x^{5} + 2\right)}{x^{5} + 2} = 4 {\color{red}\left(\frac{1}{u}\right)} \frac{d}{dx} \left(x^{3} + 4\right) + \frac{2 \frac{d}{dx} \left(x^{5} + 2\right)}{x^{5} + 2}$$

Vuelva a la variable anterior:

$$\frac{4 \frac{d}{dx} \left(x^{3} + 4\right)}{{\color{red}\left(u\right)}} + \frac{2 \frac{d}{dx} \left(x^{5} + 2\right)}{x^{5} + 2} = \frac{4 \frac{d}{dx} \left(x^{3} + 4\right)}{{\color{red}\left(x^{3} + 4\right)}} + \frac{2 \frac{d}{dx} \left(x^{5} + 2\right)}{x^{5} + 2}$$

La derivada de una suma/diferencia es la suma/diferencia de derivadas:

$$\frac{2 \frac{d}{dx} \left(x^{5} + 2\right)}{x^{5} + 2} + \frac{4 {\color{red}\left(\frac{d}{dx} \left(x^{3} + 4\right)\right)}}{x^{3} + 4} = \frac{2 \frac{d}{dx} \left(x^{5} + 2\right)}{x^{5} + 2} + \frac{4 {\color{red}\left(\frac{d}{dx} \left(x^{3}\right) + \frac{d}{dx} \left(4\right)\right)}}{x^{3} + 4}$$

Aplique la regla de potencia $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ con $$$n = 3$$$:

$$\frac{2 \frac{d}{dx} \left(x^{5} + 2\right)}{x^{5} + 2} + \frac{4 \left({\color{red}\left(\frac{d}{dx} \left(x^{3}\right)\right)} + \frac{d}{dx} \left(4\right)\right)}{x^{3} + 4} = \frac{2 \frac{d}{dx} \left(x^{5} + 2\right)}{x^{5} + 2} + \frac{4 \left({\color{red}\left(3 x^{2}\right)} + \frac{d}{dx} \left(4\right)\right)}{x^{3} + 4}$$

La derivada de una constante es $$$0$$$:

$$\frac{4 \left(3 x^{2} + {\color{red}\left(\frac{d}{dx} \left(4\right)\right)}\right)}{x^{3} + 4} + \frac{2 \frac{d}{dx} \left(x^{5} + 2\right)}{x^{5} + 2} = \frac{4 \left(3 x^{2} + {\color{red}\left(0\right)}\right)}{x^{3} + 4} + \frac{2 \frac{d}{dx} \left(x^{5} + 2\right)}{x^{5} + 2}$$

La derivada de una suma/diferencia es la suma/diferencia de derivadas:

$$\frac{12 x^{2}}{x^{3} + 4} + \frac{2 {\color{red}\left(\frac{d}{dx} \left(x^{5} + 2\right)\right)}}{x^{5} + 2} = \frac{12 x^{2}}{x^{3} + 4} + \frac{2 {\color{red}\left(\frac{d}{dx} \left(x^{5}\right) + \frac{d}{dx} \left(2\right)\right)}}{x^{5} + 2}$$

La derivada de una constante es $$$0$$$:

$$\frac{12 x^{2}}{x^{3} + 4} + \frac{2 \left({\color{red}\left(\frac{d}{dx} \left(2\right)\right)} + \frac{d}{dx} \left(x^{5}\right)\right)}{x^{5} + 2} = \frac{12 x^{2}}{x^{3} + 4} + \frac{2 \left({\color{red}\left(0\right)} + \frac{d}{dx} \left(x^{5}\right)\right)}{x^{5} + 2}$$

Aplique la regla de potencia $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ con $$$n = 5$$$:

$$\frac{12 x^{2}}{x^{3} + 4} + \frac{2 {\color{red}\left(\frac{d}{dx} \left(x^{5}\right)\right)}}{x^{5} + 2} = \frac{12 x^{2}}{x^{3} + 4} + \frac{2 {\color{red}\left(5 x^{4}\right)}}{x^{5} + 2}$$

Por lo tanto, $$$\frac{d}{dx} \left(4 \ln\left(x^{3} + 4\right) + 2 \ln\left(x^{5} + 2\right)\right) = \frac{10 x^{4}}{x^{5} + 2} + \frac{12 x^{2}}{x^{3} + 4}$$$.

Por lo tanto, $$$\frac{\frac{d}{dx} \left(H{\left(x \right)}\right)}{H{\left(x \right)}} = \frac{10 x^{4}}{x^{5} + 2} + \frac{12 x^{2}}{x^{3} + 4}$$$.

Por lo tanto, $$$\frac{d}{dx} \left(H{\left(x \right)}\right) = \left(\frac{10 x^{4}}{x^{5} + 2} + \frac{12 x^{2}}{x^{3} + 4}\right) H{\left(x \right)} = 2 x^{2} \left(x^{3} + 4\right)^{3} \left(x^{5} + 2\right) \left(11 x^{5} + 20 x^{2} + 12\right).$$$