Differentiate the following functions with respect to x: e^ ^ -1 … - Vidyadip

Question

Differentiate the following functions with respect to x:
$\text{e}^{\sin^{-1}2\text{x}}$

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Answer

Consider $\text{y}=\text{e}^{\sin^{-1}2\text{x}}$
Differentiate with respect to x,
$\frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}\big(\text{e}^{\sin^{-1}2\text{x}}\big)$
$=\text{e}^{\sin^{-1}2\text{x}}\times\frac{\text{d}}{\text{dx}}\big(\sin^{-1}2\text{x}\big)$
[Using chain rule]
$=\text{e}^{\sin^{-1}2\text{x}}\times\frac{1}{\sqrt{1-(2\text{x})^2}}\frac{\text{d}}{\text{dx}}(2\text{x})$
$=\frac{2\text{e}^{\sin^{-1}}2\text{x}}{\sqrt{1-4\text{x}^2}}$
Hence, the solution is, $\frac{\text{d}}{\text{dx}}\big(\text{e}^{\sin^{-1}}2\text{x}\big)=\frac{2\text{e}^{\sin^{-1}}2\text{x}}{\sqrt{1-4\text{x}^2}}$

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