First slide

Derivatives

Question

$Let f (x) = e^{x}, g (x) = \sin^{- 1} x and h (x) = f (g (x)) then \frac{h^{'} (x)}{h (x)}$

Moderate

A

$\sin^{- 1} x$
B

$\frac{1}{\sqrt{1 - x^{2}}}$
C

$\frac{1}{1 - x^{2}}$
D

$e^{\sin^{- 1} x}$

Solution

$h (x) = f [g (x)] = e^{\sin^{- 1} x} \Rightarrow h' (x) = e^{\sin^{- 1} x} \cdot \frac{1}{\sqrt{1 - x^{2}}} \Rightarrow \frac{h' (x)}{h (x)} = \frac{1}{\sqrt{1 - x^{2}}} .$

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