RULES FOR CALCULATING DERIVATIVES

The direct application of the definition to find the derivative of a function is typically a rather difficult and cumbersome procedure, possibly requiring some intuition.

Example 2.19 (Derivative of logarithm and exponential function) One of the most useful results concerning derivatives is that the derivative of the exponential is just the exponential itself:

As a first step to prove this deceptively simple result, it is better to find the derivative of the logarithm:

Both numerator and denominator of the increment ratio go to zero, so some manipulation is needed in order to figure out what really happens. To begin with, we may use properties of the logarithms to transform the increment ratio a bit:

If h → 0, then 1/h → ∞.¹⁵ Furthermore, the logarithm is a continuous function. Then, Eq. (2.12) applies and the limit of a logarithm is the logarithm of the limit. So

where z = 1/x. But from the basic results about Euler’s number¹⁶ we know that

which in turn implies that

This result is convincing if we look at the plot of the logarithm function in Fig. 2.15. The function is always increasing, but the rate of increase is very large for a small value of x; actually, it goes to infinity for x → 0+. The function flattens when x increases, as the rate of increase diminishes and goes to zero when x → +∞.

Now we would also like to find the derivative of the exponential function. To this aim, we need a result about the derivative of inverse functions, which is outlined below.

The procedure that we have just illustrated, although a bit informal and shaky, should convince you that we really need some handy way to find derivatives. In practice, we do the following:

1. We take advantage of basic results about the derivative of a few fundamental functions. For instance, we already know the derivative of building blocks such as the monomial xⁿ and the exponential e^x.
2. Then, we apply rules that allow to decompose the task of differentiating a complicated function into more manageable subtasks.

In this section we describe rules to find the derivative in the following cases:

• Functions obtained by summing, multiplying, or dividing other functions
• Functions obtained by composition
• Functions obtained by inversion