Differentiation (Finding Derivatives)

By M Bourne

In this Chapter

1. Limits and Differentiation
2. The Slope of a Tangent to a Curve (Numerical)
3. The Derivative from First Principles
4. Derivative as an Instantaneous Rate of Change
5. Derivatives of Polynomials
6. Derivatives of Products and Quotients
7. Differentiating Powers of a Function
8. Differentiation of Implicit Functions
9. Higher Derivatives

See also the Introduction to Calculus, where there is a brief history of calculus.

What is Differentiation?

Differentiation is all about finding rates of change of one quantity compared to another. We need differentiation when the rate of change is not constant.

What does this mean?

Constant Rate of Change

First, let's take an example of a car travelling at a constant 60 km/h. The distance-time graph would look like this:

linear graph

Revision

More on distance-time graphs.

We notice that the distance from the starting point increases at a constant rate of 60 km each hour, so after `5` hours we have travelled 300 km. We notice that the slope (gradient) is always `300/5 = 60` for the whole graph. There is a constant rate of change of the distance compared to the time. The slope is positive all the way (the graph goes up as you go left to right along the graph.)

Rate of Change that is Not Constant

Now let's throw a ball straight up in the air. Because gravity acts on the ball it slows down, then it reverses direction and starts to fall. All the time during this motion the velocity is changing. It goes from positive (when the ball is going up), slows down to zero, then becomes negative (as the ball is coming down). During the "up" phase, the ball has negative acceleration and as it falls, the acceleration is positive.

Now let's look at the graph of height (in metres) against time (in seconds).

parabolic curve

Notice this time that the slope of the graph is changing throughout the motion. At the beginning, it has a steep positive slope (indicating the large velocity we give it when we throw it). Then, as it slows, the slope get less and less until it becomes `0` (when the ball is at the highest point and the velocity is zero). Then the ball starts to fall and the slope becomes negative (corresponding to the negative velocity) and the slope becomes steeper (as the velocity increases).

TIP

The slope of a curve at a point tells us the rate of change of the quantity at that point.

parabola showing slope

Important Concept - Approximations of the Slope

Now, let's zoom in on the section of the graph near `t = 1` (where I have the rectangle in the graph above). We look at the bit between t = 0.9 s and t = 1.1 s. It looks like this:

parabola zoomed in - slope

Founders of Calculus

Isaac Newton
Sir Isaac Newton

Gottfried Leibniz
Gottfried Leibniz

See more in Introduction to Calculus.

Notice that if we zoom in close enough to a curve, it begins to look like a straight line. We can find a very good approximation to the slope of the curve at the point `t = 1` (it will be the slope of the tangent to the curve, marked in dark red) by observing the points that the curve passes through near `t = 1`. (A tangent is a line that touches the curve at one point only.)

Observing the graph, we see that it passes through `(0.9, 36.2)` and `(1.1, 42)`. So the slope of the tangent at `t = 1` is about:

`"slope"=(y_2-y_1)/(x_2-x_1)`

`=(42.0-36.2)/(1.1-0.9)`

`=5.8/0.2`

`=29\ "m"//"s"`

The units are m/s, as this is a velocity. We have found the rate of change by looking at the slope.

Clearly, if we were to zoom in closer, our curve would look even more straight and we could get an even better approximation for the slope of the curve.

This idea of "zooming in" on the graph and getting closer and closer to get a better approximation for the slope of the curve (thus giving us the rate of change) was the breakthrough that led to the development of differentiation.

Development of Differential Calculus

Up until the time of Newton and Leibniz, there was no reliable way to describe or predict this constantly changing velocity. There was a real need to understand how constantly varying quantities could be analysed and predicted. That's why they developed differential calculus, which we will learn about in the next few chapters.

Why Study Differentiation?

There are many applications of differentiation in science and engineering. You can see some of these in Applications of Differentiation.

Differentiation is also used in analysis of finance and economics.

One important application of differentiation is in the area of optimisation, which means finding the condition for a maximum (or minimum) to occur. This is important in business (cost reduction, profit increase) and engineering (maximum strength, minimum cost.)

Optimisation Example

A box with a square base is open at the top. If 64 cm2 of material is used, what is the maximum volume possible for the box?

We will return to this problem later and see how to do it in the Applications of Differentiation chapter.

The Approach We Use

The approach we follow here is the same as that discovered historically:

You can skip the first few sections if you just need the differentiation rules, but that would be a shame because you won't see why it works the way it does.

The remainder of the chapter explains how to find derivatives of more complex expressions.

We begin with a look at Limits and Differentiation »


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