Chapter 4.3, Problem 3E

To determine

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

Answer to Problem 3E

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

Explanation of Solution

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

  $y={\sin }^{-1}\left(\sqrt{2}t\right)$

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

Calculation: Using the derivation,

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

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