If f x - x -tan x and f is inverse of g- then g - x is equal toA- 1-1-gx-x2B- 1-1-gx-x2C- 1-2-gx-x2D- 1-2-gx-x2

The correct option is C Let y = f(x) ⇒ x = f^1(y) then f(x) = x + tan x ⇒ y = f^ 1 nbsp;(y) + tan (f^ 1(y)) ⇒ y = g(y) + tan (g(y)) or x = g(x) + tan (g(x)) ...

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Byju's Answer

Standard XII

Mathematics

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

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Solution

The correct option is C

Let y = f(x) $\Rightarrow$ $x$ = $f^{1}$ (y)

then $f (x)$ = $x + t a n x$

$\Rightarrow$ y = $f^{- 1} (y)$ + tan $(f^{- 1} (y))$

$\Rightarrow y$ = $g (y)$ + $t a n (g (y))$ or $x$ = $g (x)$ + $t a n (g (x))$ ....(i)

Differentiating both sides, then we get

1 = $g^{'} (x)$ + $s e c^{2} g (x) . g^{'} (x)$

$∴ g^{'} (x)$ = $\frac{1}{1 + s e c^{2} (g (x))}$ = $\frac{1}{1 + 1 + t a n^{2} (g (x))}$

= $\frac{1}{2 + (x - g (x))^{2}}$ [from Eq.(i)]

= $\frac{1}{2 + (g (x) - x)^{2}}$

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