If $f(x) = x^x\ln(5x-5)$, and $f'(x) = x^x\ln(5x-5)(g(x))$, Then What Would $g(x)$ Equal?

Question

If $f(x) = x^x\ln(5x-5)$, and $f'(x) = x^x\ln(5x-5)(g(x))$, then what would $g(x)$ equal?

So far, I have found the derivative:

• Find the derivative using the product rule, which gives $((x^x)/(x-1))+\ln(x)\ln(5x-5)+\ln(5x-5)$

• However, I am not sure how to get to $x^x\ln(5x-5)(g(x))$ from this point.

Does anyone have any ideas on how to find what g(x) would be?

Answer 1

Hint: Assuming $f(x) = x^x\ln(5x-5)$ and $f'(x) = x^x\ln(5x-5)g(x)$, then $g(x) = \frac{f'(x)}{f(x)} = (\ln f)'(x)$.