use logarithmic differentiation to find the derivative of y with respect to the given x. y =…

Explanation:

Step1: Take natural - log of both sides

$\ln y=\ln\left[(x^{5}+4)(x - 4)^{5}\right]^{\frac{1}{5}}=\frac{1}{5}\left[\ln(x^{5}+4)+5\ln(x - 4)\right]=\frac{1}{5}\ln(x^{5}+4)+\ln(x - 4)$

Step2: Differentiate both sides with respect to x

$\frac{1}{y}\frac{dy}{dx}=\frac{1}{5}\cdot\frac{5x^{4}}{x^{5}+4}+\frac{1}{x - 4}$ $\frac{1}{y}\frac{dy}{dx}=\frac{x^{4}}{x^{5}+4}+\frac{1}{x - 4}$

Step3: Solve for $\frac{dy}{dx}$

$\frac{dy}{dx}=y\left(\frac{x^{4}}{x^{5}+4}+\frac{1}{x - 4}\right)$ Substitute $y = \sqrt[5]{(x^{5}+4)(x - 4)^{5}}$ back in: $\frac{dy}{dx}=\sqrt[5]{(x^{5}+4)(x - 4)^{5}}\left(\frac{x^{4}}{x^{5}+4}+\frac{1}{x - 4}\right)$

Answer:

$\sqrt[5]{(x^{5}+4)(x - 4)^{5}}\left(\frac{x^{4}}{x^{5}+4}+\frac{1}{x - 4}\right)$