How do you find the derivative of ${(arcsin x)}^{2}$ $(arcsin x)^2$ ?

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How do you find the derivative of ${(arcsin x)}^{2}$ $(arcsin x)^2$ ?

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Answer 1 · 2017-11-18T15:18:21

Derivative is $\frac{2 arcsin x}{\sqrt{1 - x^{2}}}$ $(2arcsinx)/sqrt(1-x^2)$

Explanation:

Let us first workout derivative of $arcsin x$ $arcsinx$ . Let $y = arcsin x$ $y=arcsinx$ i.e. $sin y = x$ $siny=x$ and hence differentiating

$cos y \cdot \frac{d y}{d x} = 1$ $cosy*(dy)/(dx)=1$ or $\frac{d y}{d x} = \frac{1}{cos y} = \frac{1}{\sqrt{1 - {sin}^{2} y}} = \frac{1}{\sqrt{1 - x^{2}}}$ $(dy)/(dx)=1/cosy=1/sqrt(1-sin^2y)=1/sqrt(1-x^2)$

Hence $\frac{d}{d x} {(arcsin x)}^{2} = 2 arcsin x \times \frac{d}{d x} arcsin x$ $d/(dx)(arcsinx)^2=2arcsinx xx d/(dx)arcsinx$

= $\frac{2 arcsin x}{\sqrt{1 - x^{2}}}$ $(2arcsinx)/sqrt(1-x^2)$

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How do you find the derivative of ${(arcsin x)}^{2}$ $(arcsin x)^2$ ?

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