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NCERT Class XI Mathematics - Limits and Derivatives - Solutions

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Question : 71 of 72
Marks: +1, -0
x1+tan⁡x\frac{x}{1+\tan x}
Solution:  
Let f (x) = x1+tan⁡x\frac{x}{1+\tan x}
⇒ f (x) = x(1+tan⁡x)−1x(1+\tan x)^{-1} ... (i)
Differentiating (i) with respect to x, we get
ddx\frac{d}{dx} [f (x)] = 1 . (1+tan⁡x)−1(1+\tan x)^{-1} + x(sec⁡2x)(−1)(1+tan⁡x)−2x(\sec^2 x)(-1)(1+\tan x)^{-2}
= (1+tan⁡x)−1(1+\tan x)^{-1} - x(sec⁡2x)(1+tan⁡x)−2x(\sec^2 x)(1+\tan x)^{-2}
= 11+tan⁡x−xsec⁡2x(1+tan⁡x)2\frac{1}{1+\tan x} - \frac{x\sec^2 x}{(1+\tan x)^2}
= 1+tan⁡x−xsec⁡2x(1+tan⁡x)2\frac{1+\tan x - x\sec^2 x}{(1+\tan x)^2}
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