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JEE Main
Exam
JEE Main
Subject
Mathematics
Chapter
Differentiation

JEE Main · Mathematics · Differentiation

The derivative of f( x) w.r.t. g( x) at x=(π)/(4) , where f^ (1)=2...

The derivative of f(tan⁡x)f(\tan x) w.r.t. g(sec⁡x)g(\sec x) at x=π4x=\frac{\pi}{4}, where f′(1)=2f^{\prime}(1)=2 and g′(2)=4g^{\prime}(\sqrt{2})=4 is

Options

  1. A12\frac{1}{\sqrt{2}}
  2. B2
  3. C2\sqrt{2}
  4. D12\frac{1}{2}

Correct answer

A. 12\frac{1}{\sqrt{2}}

Step-by-step solution

ddxf(tan⁡x)ddxg(sec⁡x)=f′(tan⁡x)⋅sec⁡2xg′(sec⁡x)⋅sec⁡xtan⁡x At x=π4 we get =f′(tan⁡π4)(sec⁡2π4)g′(sec⁡π4)sec⁡π4tan⁡π4=f′(1)(2)g′(2)(2)(1)=2×242=12\begin{array}{l}\frac{\frac{d}{d x} f(\tan x)}{\frac{d}{d x} g(\sec x)}=\frac{f^{\prime}(\tan x) \cdot \sec ^{2} x}{g^{\prime}(\sec x) \cdot \sec x \tan x}\\ \text { At }x=\frac{\pi}{4} \text { we get }\\ =\frac{f^{\prime}\left(\tan \frac{\pi}{4}\right)\left(\sec ^{2} \frac{\pi}{4}\right)}{g^{\prime}\left(\sec \frac{\pi}{4}\right) \sec \frac{\pi}{4} \tan \frac{\pi}{4}}=\frac{f^{\prime}(1)(2)}{g^{\prime}(\sqrt{2})(\sqrt{2})(1)}=\frac{2 \times 2}{4 \sqrt{2}}=\frac{1}{\sqrt{2}} \end{array}
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