Practicing Success
If f(x) = x + tan x and g(x) is the inverse of f(x) then g'(x) is equal to |
$\frac{1}{1+[g(x)-x]^2}$ $\frac{1}{2-[g(x)-x]^2}$ $\frac{1}{2+[g(x)-x]^2}$ none of these |
$\frac{1}{2+[g(x)-x]^2}$ |
We have f(x) = x + tan x $\Rightarrow f\left(f^{-1}(x)\right)=f^{-1}(x)+\tan \left(f^{-1}(x)\right)$ $\Rightarrow x=g(x)+\tan (g(x)) \quad \left[∵ g(x)=f^{-1}(x)\right]$........(i) $\Rightarrow 1=g'(x)+\sec ^2(g(x)) g'(x)$ $\Rightarrow g'(x)=\frac{1}{1+\sec ^2(g(x))}$ $\Rightarrow g'(x)=\frac{1}{2+\tan ^2(g(x))}$ $\Rightarrow g'(x)=\frac{1}{2+[x-g(x)]^2}$ [Using (i)] |