Differentiation

Introduction to differentiation

We are often interested in the rate at which some variable is changing. For example, we may be interested in the rate at which the temperature is changing in a chemical reaction or in the rate at which the pressure in a vessel is changing. Rapid rates of change of a variable may indicate that a system is not operating normally and is approaching critical values.

Rates of change may be positive, zero, or negative. A positive rate of change means that the variable is increasing; a zero rate of change means that the variable is not changing; while a negative change of rate means that the variable is decreasing.

Consider the function for , shown below.

Between and , the function is decreasing rapidly. Across this interval the rate of change of the function is large and negative. Between and the function is still decreasing but not as rapidly as before. Across this interval the rate of change of the function is small and negative. There is a small interval, that the function seems to not change at all. Across that interval the rate of change is zero. Between and the function is increasing rapidly; the rate of change is large and positive.

It is often not sufficient to describe a rate of change as "large and positive" or "small and negative". A precise value is needed. The technique for calculating the rate of change of any function is called differentiation. Use of differentiation provides a precise value or expression for the rate of change of a function.

Average rate of change across an interval

We have already seen that a function can have different rates of change at different points on its graph. Let's first define and calculate the average rate of change of a function across an interval and later on we will also define the rate of change at a point. The figure below shows a function ; two possible argument values, and , and their two respective outputs and .

Consider that is increasing from to . The change in is . As increases from to , then the function increases from to . The change in is . Then the average rate of change of across the interval is

Another way to think of the average rate of change of a function is by visualising it as the slope of a line that passes through two points on the function. This line, called a secant line, can be drawn on a graph of a function so that we can quantify the value of the slope of the line. A secant line passing through the points and has a vertical rise of and a horizontal run of . The slope of the line, between the points and , is (which is exactly the same as the average rate of change).

Rate of change at a point

We often need to know the rate of change of a function at a point, and not simply an average rate of change across an interval. Let's assume that is really close to . To better reflect this is our notation, we will call what we used to call , , and what we used to call , , with being a very small number.

As mentioned earlier, the average rate of change of across the interval is

What do you think would happen if we assumed that the distance, , between the two points was made increasingly small (in Mathematics notation )?

If we assumed that, it would mean that the second point is really close to . This is exactly what we will assume in order to find the rate of change at the point . Let's say that we assumed that . If we now focus again on the graph above and assume that , the distance between the two points and would get smaller and likewise the difference between their respective outputs, and , would also get smaller. We can define those respective differences as and respectively. The term reads as "delta x" and represents a small change in the direction. In our case and .

Thus, the rate of change at a point is

Let's look at a couple of examples first and then focus on terminology and notation.

For non-linear functions a one unit increase in the value of leads to different increases in .

Terminology and notation

The process of finding the rate of change of a given function is called differentiation. The function is said to be differentiated. If (read " is equivalent to ") is a function of we say that is differentiated with respect to . The rate of change of a function is also known as the derivative of the function.

There is a notation for writing down the derivative of a function. If the function is , we denote the derivative of by

(read "dee y (by) dee x", "dee f of x dee x" and "f prime").

This is the point where you should start asking yourselves "Wait a minute, do I have to compute every time I need to find the derivative of a function at a point ?". Thankfully, the answer is no.

Table of derivatives

Table 1 lists some of the common functions used in Mathematics and Statistics and their corresponding derivatives. The symbols and are constants while the symbol represents a variable.

Ok, that is a good start but what do we do with functions like , and ?

The first function involves adding two functions (the first one being of the form while the second one is a constant function).

The second function, , involves multiplying two functions ( and ) while the last one, , involves dividing two functions ( and ).

We need to introduce some simple rules to enable us to extend the range of functions that we can differentiate.

Rules of differentiation

• Differentiation is linear: For any functions and and any real numbers and , the derivative of the function with respect to is

• Product rule: For any functions and the derivative of a function with respect to is

• Quotient rule: For any functions and the derivative of a function , where , with respect to is

• Chain rule: The derivative of the function of a composite function with respect to is What is a composite function you ask? It is a function that takes another function as its argument. So, instead of having a function that has as its input, we have a function which takes as its input. Thus, it becomes .

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