Motivation

The more things change …

Velocity is an important example of a derivative, but this is just one example. The world is full of quantities which change with respect to each other — and these rates of change can often be expressed as derivatives. It is often important to understand and predict how things will change, and so derivatives are important.

Here are some examples of derivatives, illustrating the range of topics where derivatives are found:

•Mechanics.

We saw that the derivative of position with respect to time is velocity. Also, the derivative of velocity with respect to time is acceleration. And the derivative of momentum with respect to time is the (net) force acting on an object.

•Civil engineering, topography.

Let \(h(x)\) be the height of a road, or the altitude of a mountain, as you move along a horizontal distance \(x\). The derivative \(h'(x)\) with respect to distance is the gradient of the road or mountain.

•Population growth.

Suppose a population has size \(p(t)\) at time \(t\). The derivative \(p'(t)\) with respect to time is the population growth rate. The growth rates of human, animal and cell populations are important in demography, ecology and biology, respectively.

•Economics.

In macroeconomics, the rate of change of the gross domestic product (GDP) of an economy with respect to time is known as the economic growth rate. It is often used by economists and politicians as a measure of progress.

•Mechanical engineering.

Suppose that the total amount of energy produced by an engine is \(E(t)\) at time \(t\). The derivative \(E'(t)\) of energy with respect to time is the power of the engine.

All of these examples arise from a more abstract question in mathematics:

•Mathematics.

Consider the graph of a function \(y = f(x)\), which is a curve in the plane. What is the gradient of a tangent to this graph at a point? Equivalently, what is the instantaneous rate of change of \(y\) with respect to \(x\)?

In this module, we discuss purely mathematical questions about derivatives. In the three modules Applications of differentiation, Growth and decay and Motion in a straight line , we discuss some real-world examples.

Therefore, although the Motivation section has focused on instantaneous velocity, which is an important motivating example, we now concentrate on calculating the gradient of a tangent to a curve.

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