Differentiation Question 201
Question: Let g (x) be the inverse of an invertible function $ f(x) $ which is differentiable at x = c, then $ g’(f(c)) $ equals
Options:
A) $ f’(c) $
B) $ \frac{1}{f’(c)} $
C) $ f(c) $
D) None of these
Show Answer
Answer:
Correct Answer: B
Solution:
Since $ g(x) $ is the inverse of function $ f(x) $ , therefore $ gof(x)=I(x) $ for all x. Now $ gof(x)=I(x),\ \ \forall x $
$ $
$ \Rightarrow gof(x)=x,\ \ \forall x $
$ \Rightarrow $ $ (gof)’(x)=1,\ \ \forall x $
Therefore $ g’(f(x))f’(x)=1,\ \ \forall x $
(using chain rule)
Therefore $ g’(f(x))=\frac{1}{f’(x)},\ \ \forall x\Rightarrow g’(f(c))=\frac{1}{f’(c)} $ (putting x=c)
BETA
