Knowledge organisers

Differentiation: A Level Maths knowledge organiser

Everything to know about differentiation on one page: key definitions, the formulas, a worked example, the mistakes to avoid and a checklist of what you should be able to do.

Download the A4 sheet (PDF)

Key definitions

Derivative: The gradient function dy/dx: the rate of change of y with x.
Product rule: d(uv)/dx = u dv/dx + v du/dx.
Chain rule: dy/dx = dy/du × du/dx for a function of a function.
Point of inflection: A point where the curve changes between concave and convex, so f″(x) changes sign.

Key formulas

Formulas marked with a label are given in the exam (we only say so where our formula sheet confirms it). Learn the rest.

First principlesIn the formula booklet: Edexcel, OCR	\(f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}\)
Standard	\((x^n)'=nx^{n-1},\) \((e^{kx})'=ke^{kx},\) \((\ln x)'=\tfrac1x,\) \((\sin kx)'=k\cos kx,\) \((\cos kx)'=-k\sin kx\)
Exponential base \(a\)	\((a^{kx})'=ka^{kx}\ln a\)
TrigIn the formula booklet: Edexcel, AQA, OCR	\((\tan kx)'=k\sec^2kx,\) \((\sec kx)'=k\sec kx\tan kx,\) \((\cot kx)'=-k\cosec^2kx,\) \((\cosec kx)'=-k\cosec kx\cot kx\)
Product; chain	\((uv)'=uv'+vu',\) \(\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\)
QuotientIn the formula booklet: Edexcel, AQA, OCR	\(\frac{d}{dx}\Big(\frac{f}{g}\Big)=\frac{f'g-fg'}{g^2}\)
Inverse; parametric	\(\frac{dy}{dx}=\frac1{dx/dy};\) \(\frac{dy}{dx}=\frac{dy/dt}{dx/dt}\)
Implicit	\(\tfrac{d}{dx}f(y)=f'(y)\tfrac{dy}{dx},\) \(\tfrac{d}{dx}(xy)=x\tfrac{dy}{dx}+y\)
Stationary: \(f'=0\); max if \(f''<0\), min if \(f''>0\). Convex \(f''>0\), concave \(f''<0\); inflection: \(f''\) changes sign
Tangent; normal	\(y-y_1=f'(x_1)(x-x_1);\) \(m_\text{n}=-\tfrac1{f'(x_1)}\)

Worked example

Differentiate y = x²e^(3x).

Product rule with u = x², v = e^(3x)
dy/dx = 2x e^(3x) + x² × 3e^(3x)

Answer: dy/dx = x e^(3x)(2 + 3x)

Common mistakes

H → 0 taken before simplifying
(x + h)³ expanded wrongly
Differentiating a quotient without splitting
Normal gradient not −1/m

More on what examiners see students get wrong: Examiner Insights.

You should be able to…

Estimate the gradient of a curve at a point using chords that get closer to the tangent, and see the gradient as a rate of change.
Use the derivative to find the gradient of a curve at a point and the equations of the tangent and the normal at that point.
Sketch the graph of the gradient function y = f′(x) from the graph of y = f(x), linking stationary points to the roots of f′(x).
Differentiate eˣ, e^(kx), aˣ and ln x, and use these derivatives to find gradients, tangents and stationary points of exponential and logarithmic curves.
Differentiate tan x, sec x, cosec x and cot x, and combine these results with the chain, product and quotient rules.
Use the chain rule to connect rates of change, such as finding how fast the radius of a spreading circle grows, and form simple differential equations.

The printable sheet

Differentiation knowledge organiser for A Level Maths: one A4 page of key definitions, formulas, a worked example and common mistakes — Differentiation knowledge organiser (A Level Maths), A4. Download the PDF.

Revise it next

Other A Level Maths topics: Proof · Algebra and functions · Coordinate geometry · Sequences and series · Trigonometry · Exponentials and logarithms · Integration · Numerical methods · Vectors · Sampling and data · Probability · Binomial and normal distributions · Hypothesis testing · Kinematics · Forces and Newton's laws · Moments · All A Level Maths organisers