Chapter 4.3, Problem 15E

To determine

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

Answer to Problem 15E

  $\frac{dy}{dx}=\frac{-2}{{\left(x^2+1\right)}^{\frac{3}{2}}}$

Explanation of Solution

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

  $y={\csc }^{-1}\left(x^2+1\right)$

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

Calculation: Using the derivation,

  $\begin{array}{l}\frac{dy}{dx}=\frac{d\left({\csc }^{-1}\left(x^2+1\right)\right)}{dx}\\ \frac{dy}{dx}=\frac{-1}{\left(x^2+1\right)\times \sqrt{\left({\left(x^2+1\right)}^2-1\right)}}\times \frac{d\left(x^2+1\right)}{dx}\\ \frac{dy}{dx}=\frac{-2x}{\left(x^2+1\right)\times \sqrt{\left(x^4+1+2x^2-1\right)}}\\ \frac{dy}{dx}=\frac{-2x}{\left(x^2+1\right)\times \sqrt{\left(x^2\right)\times \left(x^2+1\right)}}\\ \frac{dy}{dx}=\frac{-2x}{x\times {\left(x^2+1\right)}^{\frac{3}{2}}}\\ \frac{dy}{dx}=\frac{-2}{{\left(x^2+1\right)}^{\frac{3}{2}}}\end{array}$

So the derivative of y with respect to x is $\frac{dy}{dx}=\frac{-2}{{\left(x^2+1\right)}^{\frac{3}{2}}}$ .