Overview:

1. Gradient of a line passing through two points
2. Limits
3. Differentiation from first principles
4. Differentiating expressions of the form \(kx^n\) with respect to \(x\)
5. The gradient at a point on a curve
6. Tangents and normals

You should previously have come across the concept of gradient, as a quantity indicating the steepness of a line.

The gradient of a straight line passing through two points with coordinates \((x_1,y_1)\) and \((x_2,y_2)\) respectively is \(\dfrac{y_2-y_1}{x_2-x_1}\).

Can you see why the formula could also be written as follows? \(\text{Gradient} = \dfrac{y_1-y_2}{x_1-x_2}\)

At any, rate, do not think of this as a formula to memorise—this is just common sense!

Move the two points around in the applet below, and work out the gradient. Click the dashed lines if you need any help, and then click the “Show gradient” button to check your answer.