Chapter 4.3, Problem 20E

To determine

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

Answer to Problem 20E

  $\frac{dy}{ds}=\frac{\sqrt{\left(s^2-1\right)}}{s}$

Explanation of Solution

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

  $y=\sqrt{\left(s^2-1\right)}-{\sec }^{-1}s$

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

Calculation: Using the derivation,

  $\begin{array}{l}\frac{dy}{ds}=\frac{d\left(\sqrt{\left(s^2-1\right)}-{\sec }^{-1}s\right)}{ds}\\ \frac{dy}{dx}=\frac{1}{2\sqrt{\left(s^2-1\right)}}\times \frac{d\left(s^2-1\right)}{ds}-\frac{1}{s\times \sqrt{\left(s^2-1\right)}}\\ \frac{dy}{ds}=\frac{2s}{2\sqrt{\left(s^2-1\right)}}-\frac{1}{s\times \sqrt{\left(s^2-1\right)}}\\ \frac{dy}{ds}=\frac{s^2-1}{s\times \sqrt{\left(s^2-1\right)}}\\ \frac{dy}{ds}=\frac{\sqrt{\left(s^2-1\right)}}{s}\end{array}$

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