Worksheet for Section 3.5

Secton 3.5 is about derivative rules for trigonometric functions. Begin with the derivative of $\sin x$:

$$\frac{d}{dx}\sin x=\lim_{h\to 0}\frac{sin(x+h)-sin(x)}{h} =\sin(x)\lim_{h\to 0}\frac{\cos(h)-1}{h}+\cos(x)\lim_{h\to 0}\frac{\sin(h)}{h}$$

Explore both of these remaining limits, $$\lim_{h\to 0}\frac{\cos(h)-1}{h}$$ and $$\lim_{h\to 0}\frac{\sin(h)}{h}$$, using the Limits applet. I will sketch for you in class a more formal way to establish the second limit. The book has another approach at the end of the section. The conclusion of this, though, is a rule for the derivative of $\sin x$:

$$\frac{d}{dx}\sin x=\cos x$$

This rule can be used to establish a couple of other useful rules for derivatives of common trigonometric functions:

$$\frac{d}{dx}\cos x=-\sin x\qquad\qquad \frac{d}{dx}\tan x=\sec^2 x$$

I will show you in class how these follow from the derivative rule for $\sin x$.

Here are some example problems -- in each case, compute the derivative:

$$\frac{d}{dx}\sin(3x)\qquad \frac{d}{dx}\cos^2 x\qquad \frac{d}{dx}\cos(x^2)\qquad \frac{d}{dx}3\tan(2x)\qquad \frac{d}{dx}\tan(x-1)$$

$$\frac{d}{dx}e^{\sin x}\qquad \frac{d}{dx}[3x\sin x - 2\cos x]\qquad \frac{d}{dx}[x^2e^x-2x\cos x]\qquad \frac{d}{dx}\frac{1+\tan x}{1-\tan x}$$

In several of these, you will also have to use previous rules such as the Product Rule, Quotient Rule, and Chain Rule.