The Product and Quotient Rules, Part 2

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Anything You Want to Talk About?

Now that we’ve finished the limits part of the course, is all the grading that will affect that outcome done? No, partly because the outcome is actually “limits and derivatives,” and we have plenty more to do with derivatives, and partly because limits will come back when we define integrals.

COVID Surveillance (Pooled) Testing

If you’re asked to go give a sample during class time, do it. It’s a hugely important public health measure (the only system we have for detecting asymptomatic infections), and you can always keep up with the missed class online. I don’t penalize you for not being in class.

The Product and Quotient Rules

Based on “The Product Rule” and “The Quotient Rule” in section 3.3 of the textbook, and on the product and quotient rules discussion.

Using the Rules

Find the derivative of

There are a couple of ways you could do this, including multiplying out the numerator and using the quotient rule, treating the whole expression as a product of, say, (6t - 5) times a fraction and so using the product rule with the quotient rule inside it, or treating the expression as a fraction with a product in its numerator and so using the quotient rule with the product rule inside it. We decided to do the first of these, as follows:

What is the derivative of x² sin x (keeping in mind that the derivative of sin x is cos x)?

This uses the product rule directly:

Deriving New Rules from the Quotient Rule

Assume for the moment that the derivative of sin x is cos x, and the derivative of cos x is -sin x (both of these are true, and we’ll see why shortly).

What is the derivative of tan x?

Use the definition of tangent (sine over cosine), then use the quotient rule on that fraction, and simplify using trigonometric identities and definitions:

Can you derive a “reciprocal rule,” i.e., a rule for taking derivatives of the form 1/f(x)?

Yes, by using the quotient rule on 1/f(x). It’s a very short derivation:

Notice that this is not what you’d get if you blindly applied the power rule to f(x)^-1. That discovery suggests that there is at least one subtlety to derivatives that we don’t really know yet (it’s called the chain rule, and we’ll see it in about a week).

Next

Where the derivatives of sine and cosine come from. We’ll work up to the main results over several days, using the limit definition of the derivative. To do that, we need to take limits of expressions involving sine and cosine functions, and to do that we need another technique for finding limits.

Please read “The Squeeze Theorem” in section 2.3 of our textbook.

Use this discussion of the Squeeze Theorem to help you make sense of the reading.

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