Differentiation from first principles - part 1


Finding the gradient at a point

The tangent to a function at a point is a straight line, and so we can attempt to find the gradient of this line. You may be familiar with the following formula, or something similar, like or perhaps change in y divided by change in x However, in order to do this, we need two points on the line. As the tangent just touches the curve we only have one point. Therefore instead of a tangent we will start by considering a chord, the chord crosses the curve and so we have two coordinates and we can now calculate a gradient. But the gradient of the chord is not the same as the gradient of the tangent and so we only have an approximation to the gradient of the tangent. In order to make this approximation better and better, we will move the points on the chord closer and closer together. Let us imagine that we start with one point having an x-coordinate of 1 and a second point with an x-coordinate of 2. The difference in the x-coordinates is 1 unit. But what happens if we imagine the second point to have an x-coordinate of 1.5 or 1.1 or 1.01 or 1.001 and the difference in the x-coordinates is instead 0.5, 0.1, 0.01 or 0.001. The activity below lets you change this value (here called h) to see how the gradient of the chord changes. The gradient of the tangent is also shown for comparison


Drag the slider h and see how the gradient of the chord approaches the tangent (in red).

Question 1

What happens when h gets close to zero?

Question 2

What happens when h = 0?