CUET · MATHS · PYQ PAPER 2023
Differential coefficient of \(\sec(\tan^{-1} x)\) with respect to x is:
- A \(x\sqrt{1+x^2}\)
- B \(\frac{1}{\sqrt{1+x^2}}\)
- C \(\frac{x}{\sqrt{1+x^2}}\)
- D \(\frac{x}{1+x^2}\)
Answer & Solution
Correct Answer
(C) \(\frac{x}{\sqrt{1+x^2}}\)
Step-by-step Solution
Detailed explanation
\(\frac{d}{dx}(\sec(\tan^{-1} x)) = \sec(\tan^{-1} x)\tan(\tan^{-1} x) \cdot \frac{d}{dx}(\tan^{-1} x)\) \(= \sec(\tan^{-1} x) \cdot x \cdot \frac{1}{1+x^2}\) Let \(\theta = \tan^{-1} x\). Then \(\tan \theta = x\). From a right triangle, \(\sec \theta = \sqrt{1+x^2}\).…
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