This module uses the derivative in three ways: to write the equation of a tangent, to write the equation of a normal, and to describe how fast one quantity changes. One idea carries all of it: dy/dx is the gradient of the curve, and dy/dt is how quickly y changes with time.
Check the current Cambridge Additional Mathematics 0606 syllabus (linked below) for the exact content and notation in your exam year. Our Additional Mathematics learning guide shows where this module sits among the others.
What should you already know?
You need to differentiate powers of x, including negative and fractional powers, and use the chain rule. You also need the equation of a line, y − y₁ = m(x − x₁), and the fact that perpendicular gradients multiply to −1.
If differentiation is still slow, revise differentiation techniques first. After this module, integration methods reverse the process.
An orienting example
Find the tangent and the normal to y = x² − 4x + 5 at the point where x = 3.
Step 1, the point: y = 9 − 12 + 5 = 2, so the point is (3, 2).
Step 2, the gradient: dy/dx = 2x − 4, so at x = 3 the gradient is 2.
Step 3, the tangent: y − 2 = 2(x − 3), so y = 2x − 4.
Step 4, the normal: the gradient is −1/2. y − 2 = −(1/2)(x − 3), so 2y − 4 = −x + 3 and x + 2y − 7 = 0.
Check: 2 × (−1/2) = −1. ✓ The point (3, 2) fits both lines: 2(3) − 4 = 2 and 3 + 4 − 7 = 0. ✓
In which order should you study it?
- Find a tangent equation at a given point: the base skill, combining a point and a gradient.
- Form a normal using the correct reciprocal sign: the same gradient, flipped and negated.
- Use a derivative as a rate of change: the same derivative read as a speed of change.
- Link two changing quantities through a related-rate model: the chain rule with two varying quantities.
- Explain a rate sign in context: turning a positive or negative answer into a clear sentence.
Then work through the mixed practice set. One lesson a day and the practice set at the weekend is a steady pace.
Which traps catch most students here?
- Using the y-value as the gradient, or the gradient as the y-value.
- Half-converting the normal gradient: flipping without changing the sign, or the reverse.
- Stopping at dA/dr and forgetting to multiply by dr/dt.
- Substituting before differentiating, which turns a formula into a constant.
- Giving an average rate when the question wants the rate at one moment.
- Writing a rate without units or direction, such as “−8” with no “decreasing”.
Each lesson shows one of these slips in full and then corrects it.
How should you use the practice set?
Attempt each question on paper first, and write down the point and the gradient as two separate lines before you form an equation. Then open the worked answer and compare method as well as final value. Use the non-calculator working trainer to rehearse fraction gradients and the quadratic structure explorer to check turning points and roots.
If normal gradients or related rates keep going wrong, a teacher in online one-to-one Additional Mathematics tuition can start from the exact step where your working turns.