Derivatives of Exponential Functions

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Why does Mathematica sometimes produce derivatives that disagree with hand-calculated ones for problem set 4?

Assuming the hand-calculated ones are correct, the likely reason is small errors in Mathematica syntax.

For example Sqrt (with a capital S) is Mathematica’s square root function, but if you type sqrt (lowercase s) instead, Mathematica will think you mean some function you know about but it doesn’t. It will have no idea that your sqrt could possibly have anything to do with its Sqrt.

Also remember that function arguments always go in square brackets. So, for example, Sqrt[4] computes the square root of 4, namely 2, but Mathematica thinks that Sqrt(4) means you are for some reason trying to multiply Sqrt by 4 (and so it will dutifully answer with 4 Sqrt).

Misc

Problem Set 5

...is now available online.

The problem set focuses, though not exclusively, on trigonometric derivatives and the chain rule. See the handout for details.

To be done this week and graded late in the week or early next week.

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Derivatives of Exponential Functions

Based on “Derivative of the Exponential Function” in section 3.9 of the textbook, and the exponential derivatives discussion.

A General Rule

Can you find a general formula for the derivative of a^x, where a could be any constant? Someone correctly observed in the discussion that the derivative is (ln a) a^x, but didn’t know why. So let’s try working it out.

A good starting point is the fact that a = e^{ln a}.

Then raise both sides of that equation to the x power:

Then use one of the algebraic facts about exponents to simplify:

Now we can take the derivative, using the chain rule with e^u as the outer function and x ln a as the inner. Since ln a is a constant, multiplying it and x is just multiplying x by a constant.

Finally, use algebra on the exponents again to simplify to (ln a) a^x.

Antiderivatives

Can you “run the rule backwards” to find an antiderivative for e^x?

Can you do a similar thing for a general exponential function a^x, where a could be any constant?

The antiderivative for e^x comes straight from the rule for differentiating e^x, though the rule for a^x is a little harder. But based on what we’ve seen of exponential functions so far, we might guess that something involving a^x, maybe multiplied by some constant, would work. Trying this guess out reveals that it works if the constant is 1 / ln a:

So now we have two new antiderivative rules:

Application to Population Growth

I occasionally brew my own beer. This involves putting yeast into a mixture of water, flavorings, and lots of sugars. For a while, this is an ideal life for yeast, because all they have to do is eat the sugars, and brewers take great pains to make sure nothing that wants to eat yeast is in the solution with them. During this time, the yeast population grows exponentially, i.e, the population might be described by an equation such as P(t) = A e^ct, where P is the population of yeast and A and c are constants. For example...

Now if we’re interested in how the yeast are reproducing, we basically want the derivative of P. Nothing is killing the yeast off, so the only thing that’s changing the population is reproduction, i.e., the derivative of P is a reasonable estimate of reproduction over time:

But now compare the reproduction function to the population, and you notice that reproduction is just a constant (0.1 in this case) times the population. This suggests that in each unit of time, 1 out of every 10 yeast cells produces another (and then those new yeast start doing the same, and so forth).

We’ve done this analysis backwards from how it’s usually presented, i.e., modelers usually start with the assumption that an organism reproduces proportionally to the number already present, and show how that assumption leads to exponential growth in the population. We’ll probably be able to show how this works by the end of this course.

(Other things besides yeast reproduce exponentially, for example it’s not a bad approximate model of disease spread to assume that each infected person infects some on-average-constant number of others before recovering. That constant appears in the exponent of the population growth function, so if you want to control a spreading disease you really want to keep the constant low — this is why you’re all wearing masks, social-distancing, etc.)

Next

Most of the remaining common functions we might want to take derivatives of, e.g., logarithms, inverse trig functions, are inverses of functions we already know how to differentiate.

To find derivatives of those inverse functions, we can use something called “implicit differentiation,” which lets you find derivatives even if you don’t have an “explicit” equation of the form y = f(x) to work with.

So please read “Implicit Differentiation” in section 3.8 of the textbook, and practice with this implicit differentiation discussion, both by class time Wednesday.

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