Chapter 3, Problem 36RE

To determine

To calculate: The find the slope of the curve at $x=\pi$ .

Answer to Problem 36RE

The slope of the function $y=\sin x\cos x$ is 1.

Explanation of Solution

Given Information: The function is $y=\sin x\cos x$ .

Calculation:

The given function is,

  $y=\sin x\cos x$

The slope of a function is the rate of change of the function. That is, it is equal to the derivative of the function at x as follows:

  $\begin{array}{c}y=\sin x\cos x\\ \frac{dy}{dx}=\frac{d}{dx}\left(\sin x\cos x\right)\left(\text{Substitute the value }y\text{ in }\frac{dy}{dx}\right)\\ =\sin x\frac{d}{dx}\left(\cos x\right)+\cos x\frac{d}{dx}\left(\sin x\right)\left(\text{Apply chain rule}\right)\\ =\sin x\left(-\sin x\right)+\cos x\left(\cos x\right)\left(\frac{d}{dx}\left(\cos x\right)=-\sin x,\frac{d}{dx}\left(\sin x\right)=\cos x\right)\\ =-{\sin }^{2}x+{\cos }^{2}x\\ =\cos 2x\left(\because {\cos }^{2}x-{\sin }^{2}x=\cos 2x\right)\end{array}$

The slope at the point $x=\pi$ is,

  $\begin{array}{c}{\left[\frac{dy}{dx}\right]}_{x=\pi }={\left[\cos 2x\right]}_{x=\pi }\\ =\cos 2\pi \\ =1\left(\cos 2\pi =1\right)\end{array}$

Conclusion:

The slope of the function $y=\sin x\cos x$is 1.