Differentiation of Inverse Trigonometric Functions
If f x = x +t...
Question
If f(x) = x + tan x and f is inverse of g, then g’(x) is equal to
A
11+(g(x)−x)2
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B
11−(g(x)−x)2
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C
12+(g(x)−x)2
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D
12−(g(x)−x)2
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Solution
The correct option is C12+(g(x)−x)2 Lety=f(x)⇒x=f−1(y) thenf(x)=x+tanx ⇒x=f−1(y)+tan(f−1(y)) ⇒x=g(y)+tan(g(y))orx=g(x)+tan(g(x))⋯(i) Differentiating both sides, then we get 1=g′(x)+sec2g(x).g′(x) ∴g′(x)=11+sec2(g(x))=11+1+tan2(g(x)) =12+(x−g(x))2[fromEq.(i)] 12+(g(x)−x)2