Chapter 4.3, Problem 14E

To determine

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

Answer to Problem 14E

  $\frac{dy}{ds}=\frac{1}{s\times \sqrt{\left(25s^2-1\right)}}$

Explanation of Solution

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

  $y={\sec }^{-1}\left(5s\right)$

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

Calculation: Using the derivation,

  $\begin{array}{l}\frac{dy}{ds}=\frac{d\left({\sec }^{-1}\left(5s\right)\right)}{ds}\\ \frac{dy}{ds}=\frac{1}{5s\times \sqrt{\left({\left(5s\right)}^2-1\right)}}\times \frac{d\left(5s\right)}{ds}\\ \frac{dy}{ds}=\frac{5}{5s\times \sqrt{\left(25s^2-1\right)}}\\ \frac{dy}{ds}=\frac{1}{s\times \sqrt{\left(25s^2-1\right)}}\end{array}$

So the derivative of y with respect to x is $\frac{dy}{ds}=\frac{1}{s\times \sqrt{\left(25s^2-1\right)}}$ .