8.3 More Rules

The rules for differentiation discussed so far allow us to find formulae for derivatives of most functions you will encounter.

However there actually are other rules for differentiating that we will eventually discuss. We have not done so here because defining the functions to be differentiated involves concepts we have not yet discussed.

In particular, functions whose derivatives we have not yet considered are: infinite sums, and also areas between a function's graph and the x-axis between two given \(x\) values. Such latter things are called definite integrals. If the given x values are both finite and the function is bounded from above and below, it is called a proper integral.

The rules are quite simple when they work, so the only interesting thing about them is determining when they work.

For an infinite sum you can apply the sum rule and just sum the derivatives of all the terms to get the derivative of the sum, unless the sum becomes infinite in some way.

The same is true for differentiating a proper integral of a function with respect to a parameter that appears in that function . You can just differentiate the function with respect to that parameter and find the integral of the result, if it makes sense, and there were no infinities lurking in the problem.

You can also differentiate an integral with respect to a variable that is one of the endpoints of the area defining it. The answer then, for the upper endpoint, is the integrand, which is the function defining the integral, itself, evaluated there, as we shall see soon. (By the way, this statement is one direction of the Fundamental Theorem of Calculus and it is very easy to prove as we shall soon see. Here is the statement:

\[\frac{d}{dx} \int_{y=z}^{y=x} f(y) dy = f(x)\]

Exercise 8.5 Plot \(\tan x\) and \(\text{atan}\,x\) using the applet that takes inverses.

I am getting tired of this stuff.

Well, we really are done with what is traditionally taught about how to differentiate functions. The multiple occurrence rule, the chain rule and the inverse rule tell us how to differentiate anything we can construct, starting from the three functions \(x\), \(\exp x\), and \(\sin x\), whose derivatives we know. It is easy to make mistakes when you find derivatives, so it is wise to have a way to check your answers.


An easy way is to compare them to the results of differentiating numerically, which we next describe.