Chapter 2.5, Problem 4E

To determine

To calculate: The derivative $\frac{dy}{dx}$ of the function

Answer to Problem 4E

The derivative of the function is $\sec x\left(x\tan x+1\right)$ .

Explanation of Solution

Given information:

The function is $y=x\sec x$ .

Concept used:

The product rule is $\frac{d}{dx}\left(u.v\right)=u.\frac{dv}{dx}+v\frac{du}{dx}$ and the formulae used are $\frac{d}{dx}\left(x^n\right)=n{x}^{n-1}$ , $\frac{d}{dx}\left(\sec x\right)=\sec x\tan x$ .

Calculation:

The function is $y=x\sec x$ .

Differentiate the function with respect to $x$ .

  $\begin{array}{c}\frac{dy}{dx}=\frac{d}{dx}\left(x\sec x\right)\\ =x\frac{d}{dx}\left(\sec x\right)+\sec x\frac{d}{dx}\left(x\right)\\ =x\sec x\tan x+\sec x.1\\ =\sec x\left(x\tan x+1\right)\end{array}$

Conclusion: The derivative of the function is $\sec x\left(x\tan x+1\right)$ .