Chapter 4.3, Problem 17E

To determine

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

Answer to Problem 17E

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

Explanation of Solution

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

  $y={\sec }^{-1}\left(\frac{1}{t}\right)$

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

Calculation: Using the derivation,

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

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