Question 68 of 70intermediate🔧 ApplyShort Answer1 mark

Find the derivative of the following functions (it is to be understood that a,b,c,d,p,q,ra, b, c, d, p, q, r and ss are fixed non-zero constants and mm and nn are integers): x1+tan⁡x\frac{x}{1 + \tan x}

Correct Answer

1+tan⁡x−xsec⁡2x(1+tan⁡x)2\frac{1 + \tan x - x\sec^2 x}{(1 + \tan x)^2}

Exercise: Miscellaneous Exercise on Chapter 12 | Q: 28 | (Page Page 37)
For More Understanding

Explanation

This question requires finding the derivative using the quotient rule. The context shows the quotient rule being applied in similar problems (like finding the derivative of cot x = cos x/sin x). We apply the same method here: if f(x) = u/v, then f'(x) = (u'v - uv')/v². The derivative of tan x is sec²x as stated in Example 17 of the context.

Solution Steps

  1. Step 1: Let f(x) = x/(1 + tan x). Using the quotient rule: f'(x) = [(x)'(1 + tan x) - x(1 + tan x)']/(1 + tan x)²

  2. Step 2: The derivative of x is 1, and the derivative of (1 + tan x) is sec²x (since derivative of constant is 0 and derivative of tan x is sec²x).

  3. Step 3: Substituting: f'(x) = [1·(1 + tan x) - x·sec²x]/(1 + tan x)² = (1 + tan x - x sec²x)/(1 + tan x)²

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