Question

If $y=f(x)^{g(x)}$ and $\frac{dy}{dx} = y[H(x)f'(x)+G(x)g'(x)]$, then $\int \frac{G(x)H(x)f'(x)}{g(x)}dx =$

• A. $\log(\log f(x)) + c$
• B. $\frac{[\log f(x)]^2}{2} + c$
• C. $\frac{\log f(x)}{2} + c$
• D. $x^2+c$

Hint:

Logarithmic differentiation is the standard procedure for differentiating functions of the form $y = u(x)^{v(x)}$. The key is to take the natural log of both sides, $\ln y = v(x) \ln(u(x))$, and then use implicit differentiation and the product rule.

Answer 1

The Correct Option is B