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  4. Let f and g be differentiable functions on R such that fog is the identity function. If for some a, b ϵ R, g'(a) = 5 and g'(a)=b, then f '(b) is equal to :

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Q. Let $f$ and $g$ be differentiable functions on $R$ such that $fog$ is the identity function. If for some $a, b \,\epsilon\, R, g'\left(a\right) = 5$ and $g'\left(a\right)=b,$ then $f '\left(b\right)$ is equal to :

JEE MainJEE Main 2020Continuity and Differentiability

Solution:

$ƒ\left(g\left(x\right)\right) = x$
$ƒ'\left(g\left(x\right)\right) g'\left(x\right) = 1$
put $x = a$
$\Rightarrow ƒ'\left(b\right) g'\left(a\right) = 1$
$ƒ'\left(b\right) = \frac{1}{5}$