use logarithmic differentiation to find the derivative of y with respect to the given independent variable…

Explanation:

Step1: Take natural - log of both sides

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

Step2: Differentiate both sides with respect to (x)

Using the chain - rule, (\frac{1}{y}\frac{dy}{dx}=\frac{1}{5}\cdot\frac{5x^{4}}{x^{5}+4}+\frac{1}{x - 4})

Step3: Simplify the right - hand side

(\frac{1}{y}\frac{dy}{dx}=\frac{x^{4}}{x^{5}+4}+\frac{1}{x - 4}=\frac{x^{4}(x - 4)+(x^{5}+4)}{(x^{5}+4)(x - 4)}=\frac{x^{5}-4x^{4}+x^{5}+4}{(x^{5}+4)(x - 4)}=\frac{2x^{5}-4x^{4}+4}{(x^{5}+4)(x - 4)})

Step4: Solve for (\frac{dy}{dx})

Multiply both sides by (y=\sqrt[5]{(x^{5}+4)(x - 4)^{5}}) (\frac{dy}{dx}=\sqrt[5]{(x^{5}+4)(x - 4)^{5}}\cdot\frac{2x^{5}-4x^{4}+4}{(x^{5}+4)(x - 4)})

Answer:

(\sqrt[5]{(x^{5}+4)(x - 4)^{5}}\cdot\frac{2x^{5}-4x^{4}+4}{(x^{5}+4)(x - 4)})