TS EAMCET · Maths · Differentiation
If \(y=f(x)^{g(x)}\) and \(\frac{d y}{d x}=y\left[\mathrm{H}(x) f^{\prime}(x)+\mathrm{G}(x) g^{\prime}(x)\right]\), then \(\int \frac{\mathrm{G}(x) \mathrm{H}(x) f^{\prime}(x)}{g(x)} d x=\)
- A \(\log (\log f(x))+\mathrm{c}\)
- B \(\frac{[\log f(x)]^2}{2}+\mathrm{c}\)
- C \(\frac{\log f(x)}{2}+\mathrm{c}\)
- D \(x^2+\mathrm{c}\)
Answer & Solution
Correct Answer
(B) \(\frac{[\log f(x)]^2}{2}+\mathrm{c}\)
Step-by-step Solution
Detailed explanation
\(\log y = g(x) \log f(x)\) \(\frac{1}{y} \frac{d y}{d x} = g^{\prime}(x) \log f(x) + g(x) \frac{f^{\prime}(x)}{f(x)}\) \(\frac{d y}{d x} = y \left[ \frac{g(x)}{f(x)} f^{\prime}(x) + \log f(x) g^{\prime}(x) \right]\) Comparing with…
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