Chapter 4.3, Problem 8E

To determine

To calculate: The derivative of y with respect to x .

Answer to Problem 8E

  $\frac{dy}{dx}=\frac{-2}{\sqrt{\left(1-4x^2\right)}\times {\left({\sin }^{-1}\left(2x\right)\right)}^2}$

Explanation of Solution

Given information: The equation of y in terms of x is given as,

  $y=\frac{1}{{\sin }^{-1}\left(2x\right)}$

Formula Used: $\frac{d\left(x^n\right)}{dx}=n\times x^{\left(n-1\right)}$

Calculation: Using the derivation,

  $\begin{array}{l}\frac{dy}{dx}=\frac{d\left(\frac{1}{{\sin }^{-1}\left(2x\right)}\right)}{dx}\\ \frac{dy}{dx}=\frac{-1}{{\left({\sin }^{-1}\left(2x\right)\right)}^2}\times \frac{d\left({\sin }^{-1}\left(2x\right)\right)}{dx}\\ \frac{dy}{dx}=\frac{-1}{{\left({\sin }^{-1}\left(2x\right)\right)}^2}\times \frac{1}{\sqrt{\left(1-{\left(2x\right)}^2\right)}}\times \frac{d\left(2x\right)}{dx}\\ \frac{dy}{dx}=\frac{-2}{\sqrt{\left(1-4x^2\right)}\times {\left({\sin }^{-1}\left(2x\right)\right)}^2}\end{array}$

So the derivative of y with respect to x is $\frac{dy}{dx}=\frac{-2}{\sqrt{\left(1-4x^2\right)}\times {\left({\sin }^{-1}\left(2x\right)\right)}^2}$ .