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

Step 1: Find the derivative of $y=f(x)^{g(x)$ by logarithmic differentiation.}
Take the natural logarithm (log) of both sides: \[ \ln y = \ln(f(x)^{g(x)}) = g(x) \ln(f(x)). \] Differentiate both sides with respect to $x$, using the product rule on the right side: \[ \frac{1}{y} \frac{dy}{dx} = g'(x) \ln(f(x)) + g(x) \frac{1}{f(x)} f'(x). \] Solving for $\frac{dy}{dx}$: \[ \frac{dy}{dx} = y \left[ \left(\frac{g(x)}{f(x)}\right) f'(x) + \ln(f(x)) g'(x) \right]. \]

Step 2: Identify the functions H(x) and G(x).
We are given the form $\frac{dy}{dx} = y[H(x)f'(x)+G(x)g'(x)]$. By comparing this with the derivative we found, we can identify the corresponding functions: \[ H(x) = \frac{g(x)}{f(x)}. \] \[ G(x) = \ln(f(x)). \]

Step 3: Set up the integral with the identified functions.
We need to evaluate the integral $I = \int \frac{G(x)H(x)f'(x)}{g(x)}dx$. Substitute the expressions for $G(x)$ and $H(x)$: \[ I = \int \frac{\ln(f(x)) \cdot \frac{g(x)}{f(x)} \cdot f'(x)}{g(x)} dx. \]

Step 4: Simplify the integrand and perform the integration.
The $g(x)$ term cancels from the numerator and denominator: \[ I = \int \ln(f(x)) \cdot \frac{f'(x)}{f(x)} dx. \] This integral is in a form suitable for substitution. Let $u = \ln(f(x))$. Then $du = \frac{1}{f(x)} \cdot f'(x) dx$. The integral becomes: \[ I = \int u \, du = \frac{u^2}{2} + c. \] Substituting back for $u$: \[ I = \frac{[\ln(f(x))]^2}{2} + c = \frac{[\log f(x)]^2}{2} + c. \]