Question

Differentiate the following functions by proper substitution
sin-1 2x√1-x^2
​

Answer 1

Given ,

The function is $\sf f(x) = {sin}^{ - 1} 2x \sqrt{1 - {(x)}^{2} }$

Let , x = Sin(Φ) => Sin^-1x = Φ , so

$\sf \Rightarrow f(x) = {sin}^{ - 1} 2sin (\theta) \sqrt{1 - {(sin )}^{2}(\theta) } \\ \\\sf \Rightarrow f(x) = {sin}^{ - 1} 2sin (\theta) \sqrt{ {(cos )}^{2}(\theta) } \\ \\\sf \Rightarrow f(x) = {sin}^{ - 1} 2sin (\theta)cos(\theta) \\ \\\sf \Rightarrow f(x) = {sin}^{ - 1} sin(2\theta) \\ \\\sf \Rightarrow f(x) = 2(\theta) \\ \\\sf \Rightarrow f(x) = 2 {sin}^{ - 1}x$

Now , Differentiating with respect to x , we get

$\sf \Rightarrow \frac{dy}{dx} = \frac{d(2 {sin}^{ - 1}x )}{dx} \\ \\ \sf \Rightarrow \frac{dy}{dx} = \frac{2}{ \sqrt{1 - {(x)}^{2} } }$

$\therefore \sf \bold{ \underline{The \: derivative \: of \: given \: function \: is \: \frac{2}{ \sqrt{1 - {(x)}^{2} } } \: \: \: \: }}$