First slide

Derivatives

Question

$Let f : R \to R be a one-one onto differentiable function such that f (2) = 1 and f^{1} (2) = 3 then find the value of {(\frac{d}{d x} (f^{- 1} (x)))}_{x = 1}$

Moderate

A

1
B

$\frac{1}{2}$
C

$\frac{1}{3}$
D

$\frac{1}{4}$

Solution

$\begin{array}{l} let f^{- 1} (x) = g (x) \\ f (g (x)) = x \Rightarrow f^{'} (g (x)) g^{'} (x) = 1 \\ g^{'} (x) = \frac{1}{f^{'} (g (x))} \\ g^{'} (1) = \frac{1}{f^{'} (g (1))} = \frac{1}{f^{'} (2)} = \frac{1}{3} \end{array}$

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