Diffrentiate the following w. r. t. x. ^ -1 (x^ 3 2 ) - Vidyadip

Question

Diffrentiate the following w. r. t. x.
$\sin ^{-1}\left(x^{\frac{3}{2}}\right)$

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Answer

Let $y =\sin ^{-1}\left(x^{\frac{3}{2}}\right)$Differentiating w.r.t. x, we get
$\frac{d y}{d x}=\frac{d}{d x}\left[\sin ^{-1}\left(x^{\frac{3}{2}}\right)\right]$
$=\frac{1}{\sqrt{1-\left(x^{\frac{3}{2}}\right)^2}} \cdot \frac{d}{d x}\left(x^{\frac{3}{2}}\right)$
$=\frac{1}{\sqrt{1-x^3}} \times \frac{3}{2} x^{\frac{1}{2}}$
$=\frac{3 \sqrt{x}}{2 \sqrt{1-x^3}} .$

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