TS EAMCET · Maths · Differentiation
Let \(g(x)\) be the anti-derivative of \(f(x)\). Then the function for which \(\log _e\left(1+(g(x))^2\right)+c\) is an anti-derivative is]
- A \(\left(1+(g(x))^2\right) g^{\prime}(x) f(x)\)
- B \(\frac{-2 f(x) g(x)}{1+g(x)}\)
- C \(\frac{2 f(x) g(x)}{1+(g(x))^2}\)
- D \(\frac{2 g(x)}{1+(g(x))^2}\)
Answer & Solution
Correct Answer
(C) \(\frac{2 f(x) g(x)}{1+(g(x))^2}\)
Step-by-step Solution
Detailed explanation
\( \frac{d}{dx} \left( \log_e\left(1+(g(x))^2\right)+c \right) = \frac{1}{1+(g(x))^2} \cdot \frac{d}{dx}\left(1+(g(x))^2\right) \) \( = \frac{1}{1+(g(x))^2} \cdot (2g(x)g'(x)) \) Since \(g'(x) = f(x)\): \( = \frac{2g(x)f(x)}{1+(g(x))^2} \)
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