SUNY Geneseo Department of Mathematics

Trignometric Derivatives, Part 2

Friday, February 22

Math 221 03
Spring 2019
Prof. Doug Baldwin

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Previous Lecture

Misc

SI session this afternoon (3:00, Bailey 209)

Questions?

Trigonometric Derivatives

Section 3.5.

Functions other than Sine and Cosine

What’s the derivative of sec x?

This is an example of how you can differentiate trig functions other than sine and cosine by defining the function in terms of sine and cosine and then using the quotient rule.

secant of x equals 1 over cosine x

Specifically...

Use the quotient rule to differentiate 1 over cosine x

From here, you can rearrange terms to simplify:

Quotient rule applied to 1 over cosine yields sine over cosine squared aka tangent times secant

Practice

Find dy/dx if y = 3sinx - 2cosx

This mostly draws on old rules (the constant multiple rule, difference rule), plus the derivatives of sine and cosine.

Derivative of 3 sine x minus 2 cosine x is 3 cosine x plus 2 sine x

How about the antiderivative?

Start with some antiderivative rules very analogous to differentiation rules, e.g., constant multiple, difference:

Integral of 3 sine x minus 2 cosine x is 3 times integral of sine x minus 2 times integral of cosine x

Now we have to figure out what the antiderivatives of sine and cosine are. Cosine is relatively easy, since we already know it’s the derivative of sine, so sine (+ a constant) is the antiderivative of cosine. For the antiderivative of sine, we can reason that if negative sine is the derivative of cosine, then maybe sine is the derivative of negative cosine. And of course we can check this guess and find that it’s right:

Integral of sine x is minus cosine x, integral of cosine is sine

With these ideas, finding the rest of the antiderivative is mostly a matter of plugging them in and simplifying:

3 times integral of sine x minus 2 times integral of cosine x is -3 cosine x minus 2 sine x plus a constant

Find f′(x) if f(x) = secx tanx

This is mainly a matter of using the product rule for derivatives:

Product rule for derivative of secant times tangent yields secant times tangent squared plus secant cubed

But it simplifies further:

Secant times tangent squared plus secant cubed becomes sine squared plus 1 all over cosine cubed

Next

Nothing we’ve done so far would allow you to, for example, differentiate sin(3x).

Derivatives of functions applied to other functions require the so-called “chain rule.”

Read about it, and think about how you’d differentiate sin(3x).

Section 3.6.

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