Differentiation techniques

Differentiation gives you the gradient of a curve at any point, as a formula. In Additional Mathematics, the technique questions ask you to produce that formula correctly from powers, products, brackets raised to a power, and ln or exponential forms.

Later topics depend on it. Stationary points and optimisation and tangents, normals and rates both begin with a derivative, so an error here carries straight through. Confirm the exact list of functions you must differentiate on the current 0606 syllabus page before you revise.

What should you know before starting?

You need fluent indices, because every term must be written as a power of x first. That includes negative and fractional powers, such as 1/x² = x⁻² and √x = x^1/2.

You also need basic function notation and the idea of a gradient as “rise over run”. If composite functions still feel shaky, look at functions and restrictions. For ln and e^x, see exponential and logarithmic reasoning.

An orienting worked example

Find the gradient of y = x⁴ − 2/x at x = 1.

Step 1, rewrite as powers: y = x⁴ − 2x⁻¹.

Step 2, differentiate each term: the power comes down and the index drops by one. dy/dx = 4x³ + 2x⁻² = 4x³ + 2/x².

Step 3, substitute: at x = 1 the gradient is 4 + 2 = 6.

The two sign changes in step 2 are where marks are lost. Bringing −1 down and multiplying by −2 gives +2.

In what order should you study the lessons?

1. Differentiate a polynomial with fractional powers: the base skill. Every later rule ends with this step.
2. Differentiate a product where applicable: when two brackets multiply, and when expanding is quicker.
3. Apply a chain rule to a composite expression: for a bracket or root raised to a power.
4. Differentiate logarithmic and exponential forms where in scope: adds ln and e^x, and combines them with earlier rules.
5. Check a derivative by comparing local gradients: a habit that catches wrong answers without a mark scheme.

Then try the differentiation techniques practice set. The calculus shape and rate explorer lets you see a curve and its gradient together.

What are the common traps?

• Forgetting the inner derivative in a chain rule question.
• Multiplying the two derivatives in a product instead of using u′v + uv′.
• Leaving 1/x² or √x unrewritten and then guessing a rule.
• Mishandling negative indices, so x⁻² becomes x⁻¹ instead of x⁻³ after differentiating.
• Forgetting that ln(ax + b) needs a factor a in the numerator.

How should you use the practice set?

Attempt each question on paper first. Write the rule into your first line, such as “let u = … and v = …”, so that your method is visible.

Afterwards, check two or three answers with the local gradient habit from the last lesson. The “If you got these wrong” section at the end of the practice set points each error type back to one lesson.

If method choice is the problem rather than the rules, our online one-to-one Additional Mathematics tuition can focus on exactly that, and the wider Additional Mathematics learning guide shows where this topic sits.