Question

question 14 · 1 point. if f(x) = \frac{x^{4}e^{x}(x - 1)}{}, find f(x) using logarithmic differentiation. select the correct answer below: o f(x)=\frac{e^{x}(x - 1)}{x^{8}}(\frac{x^{2}-x}{x^{2}-8x + 8}) o f(x)=\frac{e^{x}(x - 1)}{x^{8}}(\frac{x^{2}-8x + 8}{x - x^{2}}) o f(x)=\frac{x^{2}-8x + 8}{x^{2}-x} o f(x)=\frac{e^{x}(x - 1)}{x^{8}}(\frac{x^{2}-8x + 8}{x^{2}-x})

Explanation:

Step1: Take natural - log of f(x)

$\ln(f(x))=\ln(x^{4})+\ln(e^{x})-\ln(x - 1)$

Step2: Differentiate both sides

$\frac{f'(x)}{f(x)}=\frac{4}{x}+1-\frac{1}{x - 1}$

Step3: Solve for f'(x)

$f'(x)=f(x)(\frac{4}{x}+1-\frac{1}{x - 1})=\frac{e^{x}x^{4}}{x - 1}(\frac{4(x - 1)+x(x - 1)-x}{x(x - 1)})=\frac{e^{x}(x - 1)}{x^{8}}(\frac{x^{2}-8x + 8}{x^{2}-x})$

Answer:

The first option: $f'(x)=\frac{e^{x}(x - 1)}{x^{8}}(\frac{x^{2}-8x + 8}{x^{2}-x})$