Section 2.5

You are responsible for not only memorizing the derivatives of all 6 trig functions (see formulas 3-8 in the blue boxes) but proving them if asked on a test. The proof of the derivative of sine is given above formula 3. The proof of the derivative of cosine is similar. The proof of the derivative of tangent is given below formula 8. A similar proof works for the other three trig functions.

For problems like 17 and 18 that involve both a product and a quotient, you will have to use both the product rule and the quotient rule. To help you determine which rule to start with, try the following. Plug in a number for x, any number you want. Follow the order of operations as if you were trying to calculate the corresponding y. If the last operation you perform is a multiplication, then it is a product and you should start with the product rule. If the last operation you perform is a division, then it is a quotient and you should start with the quotient rule.

For #29, remember that horizontal lines have slopes of zero. Thus finding the points where f has a horizontal tangent line is the same as finding the points where the slope of the tangent line (i.e. the derivative) is zero. Set the derivative equal to zero and solve for x.

For problem #43, try instead to find where f is not differentiable. Then answer that f is differentiable everywhere except ……. Then list the points where the derivative of f is undefined. There is a subtle difference between finding points of discontinuity and finding points of nondifferentiability. A function is discontinuous where the denominator of f(x) is zero. A function is not differentiable where the denominator of is zero.