Chapter 4.3, Problem 18E

To determine

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

Answer to Problem 18E

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

Explanation of Solution

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

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

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

Calculation: Using the derivation,

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

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