The Gradient of a Straight Line

The differential calculus is concerned with gradients, which can be thought of as representing the rate of change of one quantity with respect to another.  Consider the straight line in the following diagram.  The equation of this line takes the general form: 

where m and c are constants. 

The gradient of this straight line is defined as the increase in the value of y for a unit increase in x.  It is therefore equal to the change in y divided by the corresponding change in x, and so is  given by  

.                                     (1)

The gradient of the straight line is therefore a constant – a given increase in x gives rise to the same increase in f, regardless of the value of x.

The Gradient of a Curve 

Now consider the curve shown in the following diagram: 

In the previous diagram showing the straight line, we paid no regard to the relative values of x and x₁ when defining the gradient of the line.  We can take any two values and still obtain the correct gradient.

However, a difficulty arises because a gradient can be defined at any point on a line or curve.  For the straight line on the first figure, the gradient is the same at all points on the line, which is why we can be casual about the definition of the gradient. 

However, on curves such as in the above diagram, the gradient varies at different points on the curve.  The gradient at a point on the curve is simply the gradient of the straight line that is tangent to the curve at that point.  For example, in the diagram above, the gradient at x is simply the gradient of the tangent line labelled “gradient line at x”.

(Expert mathematicians will note that there are some types of curve for which defining the gradient at a point is more difficult.  Examples are functions that have discontinuities.  We are not concerned with such complexities here.) 

It can be seen from the above diagram that if x and x₁ are, in some sense, not “close” to each other, then the estimate of gradient obtained by using equation (1) may not be close to the true gradient.  (To see this, compare the approximate gradient and the true gradient line in the above diagram). 

If we are to use an expression such as (1) to define the gradient at a point on the curve, then we must choose an x₁ that is sufficiently close to x, and this leads to the idea of a derivative defined by 

                                      (2)

It can be seen that equation (2) is very similar to equation (1), except that we have now set

and we are letting .  Equation (2) is the gradient of the gradient line at x.

Why do we let ?  If you consider the figure above, then you can see that if we let x₁ approach the value of x, then the approximate gradient line gets closer to the true gradient line.  As the approximate gradient line becomes coincident with the true gradient line.

Example calculation 

A simple example will illustrate the computation of derivatives.  Consider the function 

The derivative of y with respect to x is then given by 

Important Rules for Derivatives

Finally, here are some important rules concerning derivatives that are used in the mathematical problems in this series of articles: 

For example: 

Another important rule is the chain rule of differentiation.  Suppose we have a function y that is a function of a variable u, and the variable u is itself a function of another independent variable x.  Then we have 

For example: 

.