Differentiation measures how fast one quantity changes compared with another. On a graph it gives the gradient of a curve at a point, and in an exam it is used to find tangents, turning points and best-value answers.
This page lists the lessons in a study order and shows how they connect. It belongs to SPM Additional Mathematics and its Form 5 route.
In what order should you learn it?
- Differentiating powers, products and quotients gives you the basic rules.
- Applying the chain rule handles brackets raised to a power.
- Finding tangents and normals uses the derivative as a gradient.
- Finding stationary points and testing their nature sets the derivative to zero.
- Solving optimisation and rates-of-change problems turns words into a function first.
- Using small changes and approximations estimates a change without recalculating.
Each lesson has a worked example and a self-check with an explained answer.
What does one curve show?
Picture this example. Take y = x³ − 3x.
The derivative is dy/dx = 3x² − 3. At x = 2 the gradient is 3(4) − 3 = 9, so a tangent there is steep. Setting 3x² − 3 = 0 gives x = 1 or x = −1, the stationary points, where y = −2 and y = 2.
A small step from x = 2 to x = 2.01 changes y by about 9 × 0.01 = 0.09. One function has given you a gradient, a turning point and an estimate, and every lesson above is one of those uses.
Where should you start?
- Fresh to differentiation: begin with powers, products and quotients and use the polynomial differentiation tutor to drill the rules.
- Rules fine, word problems weak: go to optimisation and rates of change, then try recovering the method in a mixed calculus problem.
- Close to a test: work through the differentiation practice set and note which question types cost marks.
Differentiation and integration undo each other, so studying one makes the other easier.
When is tuition worth considering?
The free lessons show the method. A one-to-one teacher watches you meet a question you have not seen and shows where to begin.
Read online one-to-one Additional Mathematics tuition to see how a lesson runs.