Differentiation Question 224
Question: Let $ g(x) $ be the inverse of the function $ f(x) $ and $ f’(x)=\frac{1}{1+x^{3}} $ . Then $ {g}’(x) $ is equal to
[Kurukshetra CEE 1996]
Options:
A) $ \frac{1}{1+{{(g(x))}^{3}}} $
B) $ \frac{1}{1+{{(f(x))}^{3}}} $
C) $ 1+{{(g(x))}^{3}} $
D) $ 1+{{(f(x))}^{3}} $
Show Answer
Answer:
Correct Answer: C
Solution:
Since $ g(x) $ is the inverse of $ f(x) $ , therefore $ g(y)=x $
$ \Leftrightarrow g^{-1}(y)=x $
Now, $ g’(f(x))=\frac{1}{f’(x)},\forall x \in D $
Therefore $ g’(f(x))=1+x^{2},\ \ \forall x $
Therefore $ g’(y)=1+{{(g(y))}^{3}} $
[using $ f(x)=y\Leftrightarrow x=g(y)] $
Therefore $ g’(x)=1+{{(g(x))}^{3}} $ (replacing y by x).
BETA
