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4th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781337687805

Chapter 3.4, Problem 48E

(a)

To determine

**To sketch:** The slope of the function

Expert Solution

The value of the derivative of the function at any point *x* can be estimated by drawing the tangent line at any point

Mark the slope of the tangent as a point in *y*-axis and the value of *x* as a point in *x*-axis in the graph of

Proceed in the similar way at several points and obtain the rough graph of the

**Graph:**

The rough sketch of the derivative function,

(b)

To determine

**To calculate:** The derivative of function

Expert Solution

The derivative of

**Result used: Chain Rule**

If *h* is differentiable at *x* and *g* is differentiable at *x* and

**Calculation:**

Obtain the derivative.

Let

Apply the chain rule as shown in equation (1),

The derivative of

Substitute

Thus, the derivative is

The derivative of

Thus the derivative is

Substitute

Therefore, the derivative of

**Graph:**

Use the online graphing calculator to draw the graph of the derivative function

From Figure 2, it is observed that the given trigonometric function is differentiable everywhere and the rough sketch of the derivative