A Level maths differentiation basics begin with a simple idea: the derivative tells you the gradient of a curve at a point. Once you understand that meaning, the power rule and tangent-line questions become far easier to learn.
A Level maths differentiation basics: what to know
This practical guide explains A Level maths differentiation basics through examples, checks and realistic next steps for students and parents.
What differentiation tells us
A straight line has a constant gradient. A curve changes steepness, so differentiation gives a formula for the gradient at any chosen x-value. The derivative of y = f(x) is commonly written f′(x) or dy/dx.
The power rule
For y = xⁿ, the derivative is dy/dx = nxⁿ⁻¹. For example, the derivative of x³ is 3x², while the derivative of 5x² is 10x. Constants differentiate to zero.
Worked example 1: differentiate a polynomial
For y = 3x³ − 4x² + 7x − 5, differentiate each term: dy/dx = 9x² − 8x + 7. Keep the signs carefully.
Worked example 2: find a gradient
For y = x² + 2x, the derivative is 2x + 2. At x = 3, gradient = 2(3) + 2 = 8. This is the gradient of the tangent at x = 3.
Worked example 3: equation of a tangent
For y = x² at x = 2, the point is (2,4). The derivative is 2x, so gradient = 4. Using y − y₁ = m(x − x₁), the tangent is y − 4 = 4(x − 2), giving y = 4x − 4.
Mini-quiz with answers
- Differentiate y = x⁵.
- Differentiate y = 6x² − 3x + 9.
- Find the gradient of y = x³ at x = 2.
Answers: 5x⁴; 12x − 3; 12. Check the derivative first, then substitute the given x-value.
Build confidence with exam questions
Once the power rule is secure, practise mixed questions involving tangents, stationary points and applications as required by the examination board. See Edexcel A Level past papers or explore A Level Maths tutoring in Abu Dhabi for one-to-one guidance.
Questions parents and students ask
What is the first rule to learn in A Level differentiation?
The power rule is a common starting point for polynomial functions.
What does f prime mean?
f′(x) is notation for the derivative, which gives the gradient function.
Is differentiation the same in every A Level course?
Core principles overlap, but the depth and sequence vary by examination board and qualification.