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

Question

Differentiate the following functions with respect to x:
$\big(\sin^{-1}\text{x}^4\big)^4$

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Answer

Consider $\text{y}=\big(\sin^{-1}\text{x}^4\big)^4$
Differentiate it with respect to x,
$\frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}\big(\sin^{-1}\text{x}^4\big)^4$
$=4\big(\sin^{-1}\text{x}^4\big)\frac{\text{d}}{\text{dx}}\big(\sin^{-1}\text{x}^4\big)$
[Using chain rule]
$=4\big(\sin^{-1}\text{x}^4\big)^3\frac{1}{\sqrt{1-\big(\text{x}^4\big)^2}}\frac{\text{d}}{\text{dx}}\big(\text{x}^4\big)$
$=4\big(\sin^{-1}\text{x}^4\big)^3\frac{4\text{x}^3}{\sqrt{1-\text{x}^8}}$
$=\frac{16\text{x}^3\big(\sin^{-1}\text{x}^4\big)^3}{\sqrt{1-\text{x}^8}}$
Hence, the solution is, $\frac{\text{d}}{\text{dx}}\big(\sin^{-1}\text{x}^4\big)=\frac{16\text{x}^3\big(\sin^{-1}\text{x}^4\big)^3}{\sqrt{1-\text{x}^8}}$

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