SUNY Geneseo Department of Mathematics

Trigonometric Derivatives and Antiderivatives

Friday, September 27

Math 221 06
Fall 2019
Prof. Doug Baldwin

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Previous Lecture

Questions?

Proving Difference Rule?

For the question in problem set 4 that asks you to prove the difference rule, I’m envisioning answers that start with a function that is the difference of two others, and the limit definition of derivative, and that end with the difference rule:

From limit definition of derivative of g minus k, series of steps leads to g prime minus k prime

Mathematica?

Mathematica may give different answers from what you get when calculating derivatives, but those answers should be simplifiable to yours, either by Mathematica or by hand. For example...

Use Simplify to make a result from D more recognizable

Trigonometric Derivatives

As an example of trigonometric differentiation rules, try finding the derivative of 2 cos x - cot x sec x.

There are at least 2 ways to go about this. One rewrites cotangent and cosecant in terms of sine and cosine and then differentiates:

Rewrite cotangent times secant and differentiate

The other uses the product rule to differentiate the “cot x sec x” term and then simplifies. (This uses slightly different trig identities than we used in class in order to give a shorter derivation, but is similar in exercising a number of trig definitions and algebra techniques.)

Use product rule to differentiate cotangent times secant, then simplify via identity and algebra

Key points from both solutions are using trigonometric differentiation rules, and how those rules combine with previous rules for sums, differences, products, etc.

Trigonometric Antiderivatives

Key antiderivative rules that follow from the differentiation rules we’ve found:

Antiderivative of sine is negative cosine, of cosine is sine

As you see more antidifferentiation rules, it’s helpful to know that they are collected in tables at the end of nearly every calculus book (including ours). You need to remember where to look them up more than remember what all the rules are.

The same is true of differentiation rules.

Next

The chain rule.

Read section 3.6.

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