Question

question find the derivative of the function $y=(x - 2)^{4x}$. select the correct answer below: $\frac{dy}{dx}=-4ln(x - 2)+\frac{(x - 2)^{4x}}{x - 2}$, $\frac{dy}{dx}=(x - 2)^{4x}(4ln(x - 2)+\frac{4x}{x - 2})$, $\frac{dy}{dx}=4x(x - 2)^{4x}+\frac{(x - 2)^{4x}}{x - 2}$, $\frac{dy}{dx}=(x - 2)^{4x}(4xln(x - 2)+\frac{4}{x - 2})$

Explanation:

Step1: Take natural - log of both sides

$\ln y = 4x\ln(x - 2)$

Step2: Differentiate both sides

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

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

$\frac{dy}{dx}=(x - 2)^{4x}(4\ln(x - 2)+\frac{4x}{x - 2})$

Answer:

$\frac{dy}{dx}=(x - 2)^{4x}(4\ln(x - 2)+\frac{4x}{x - 2})$