Mastering Differentiation in A-Level Maths

Introduction - Mastering Differentiation in A-Level Maths

There are just some topics that make up a course. Differentiation at A Level Maths is such a topic. It is everywhere. Calculus, parametric equations and also mechanics. Year 12 differentiation you are just generally using the standard rule. In year 13, you meet a set of new rules to expand your current knowledge. In this article you will understand differentiation so much better.

🔙 Previous topic:

“Revisit integration methods before diving deeper into differentiation.”

Just What is Differentiation?

In simple words differentiation is all about finding how things change. It is the process of finding the rate of change of a function. Differentiation tells you how fast a function is either increasing or decreasing.

For example, if $y = x^2.$ , then the first derivative is $\boxed{\frac{dy}{dx} = 2x.}$ which gives the gradient function of the curve at any point.

You will use differentiation in a number of areas:

Finding gradients of curves.
Determining the nature of turning points of a curve.
Calculating motion in mechanics where the acceleration is variable.

Working with exponential and logarithmic growth.

Basic Rules of Differentiation

There are some key rules to recall:

Power rule
$\frac{d}{dx}\bigl[x^n\bigr] = n\,x^{n-1}.$

Constant multiple rule
$\frac{d}{dx}\bigl[c\,f(x)\bigr] \;=\; c\;\frac{d}{dx}\bigl[f(x)\bigr].$

Sum and difference rule

Sum rule states: Sum rule states: $\frac{d}{dx}\bigl[f(x) + g(x)\bigr]= \frac{d}{dx}\bigl[f(x)\bigr]+ \frac{d}{dx}\bigl[g(x)\bigr],$

Difference rule states: Difference rule states: $\frac{d}{dx}\bigl[f(x) – g(x)\bigr]= \frac{d}{dx}\bigl[f(x)\bigr]- \frac{d}{dx}\bigl[g(x)\bigr].$

These rules set the foundation. Although you only really need to know the power rule. The other rules are extensions. .

The Chain Rule

Suppose you want to differentiate $y = (3x+2)^5$ . Here you need to use the chain rule. This is for differentiating a function of a function.

The outer function is $u^5$
The inner function is $u = 3x + 2$

The derivative is :
$\boxed{\frac{dy}{dx} = 15\,(3x + 2)^4.}$

You can use the chain rule to areas of trigonometry, exponentials, and logarithmic functions with a function inside.

Product and Quotient Rules

Sometimes you might need to differentiate a function that is multiplied with another function. You may even need to differentiate a function that is divided by another function.

Then you need to either use the product or quotient rule.

Product rule:
$\frac{d}{dx}\bigl[f(x)\,g(x)\bigr] = f'(x)\,g(x) \;+\; f(x)\,g'(x).$

Quotient rule:
$\frac{d}{dx}\!\biggl[\frac{f(x)}{g(x)}\biggr] = frac{f'(x)\,g(x)\;-\;f(x)\,g'(x)}{\bigl[g(x)\bigr]^{2}}.$

Exam Tip

If a question wants you to find the turning points then you need to simplify your derivative. This will make determining the coordinates of the turning points much easier. If it this is not asked, then there is no need to simplify.

Implicit Differentiation

You are not always going to see y written in terms of x. Sometimes you will see that y is mixed with x in an equation.

Example: $x^2 + y^2 = 25$

Differentiate both sides with respect to x:
$2x + 2y\,\frac{dy}{dx} = 0.$

Making the first derivative the subject will give:

$2y\,\frac{dy}{dx} = -2x\quad\Longrightarrow\quad\frac{dy}{dx}= \frac{-2x}{2y}= -\frac{x}{y}.$

Implicit differentiation is very common. You must know how to differentiate y in terms of x. Implicit differentiation applies to circle, ellipse, and related rates problems.

Logarithmic Differentiation

If you are dealing with something very complicated then sometimes it is much easier to logarithms on both sides. Logarithmic differentiation helps with complicated functions.

Example: $y = x^x$

Take logs: $\ln y = \ln\bigl(x^x\bigr).$

Differentiate: $\frac{d}{dx}\bigl(\ln y\bigr) = \frac{d}{dx}\bigl(x\,\ln x\bigr).$

Multiply by y: $\frac{1}{y}\,\frac{dy}{dx} = \ln x + 1.$

You can use this technique for products, quotients, and powers with variables in the exponent.

Stationary Points

Stationary points are also known as turning points. They happen when Stationary points occur when $\displaystyle \frac{dy}{dx}=0$ .

These points can be:

Maximum – the gradient of the curve goes from increasing to decreasing
Minimum – the gradient of the curve goes from decreasing to increasing

Point of inflection – the gradient of the curve does not change either side of the turning point

You can use the second derivative test to quickly determine the nature of a turning point:

$\displaystyle \frac{d^2y}{dx^2}(c)>0$ → you have a minimum turning point

$\displaystyle \frac{d^2y}{dx^2}(c)<0$ → you have a maximum turning point

$\displaystyle \frac{d^2y}{dx^2}(c)<0$ → you have a maximum turning point $\displaystyle \frac{d^2y}{dx^2}(c)=0$ → then you must test the gradient either side of the turning point

Tangents and Normals

When you have the gradient function and substitute a value for x, this provides the gradient of the tangent of the curve at that particular point.

Equation of tangent:
$y – y_{1} = m\bigl(x – x_{1}\bigr)$

Where m is the gradient
Where $x_{1}$ and $y_{1}$ are specific points on the curve.

Remember that a line normal to the tangent is perpendicular to it. Two lines are perpendicular when $m_{1}\,m_{2} = -1.$ .

Tips for Mastering Differentiation

Practice – differentiation improves with regular practice.
Know the key rules – chain, product, quotient, exponential, and logarithm rules.
Show all working– examiners want to see what you know. Do not take short cuts
Check your units in mechanics – this avoids mistakes.
Always draw diagrams for questions involving tangents, normals, and stationary points.

Differentiation is tested in all of the A-Level exam boards (AQA, Edexcel, OCR, MEI). Knowing differentiation is a big part of the content and will help with calculus, mechanics, and statistics.

Ready to boost your exam confidence?

If you are looking for more indepth help regarding exams, exam technique and tackling harder exam questions explore our half term 3 Day A Level Maths Revision Course which takes place online.

Author Bio – S. Mahandru

S. Mahandru • Head of Maths, Exam.tips

S. Mahandru has over 15 years’ experience teaching A-Level Maths in UK classrooms. He is a published author and an approved A-Level examiner. He also specialises in calculus, mechanics, and statistics, helping students boost grades and confidence.

🧭 Next topic:

“Now explore how to recognise common differentiation question types.”

FAQS

What are common mistakes students make with differentiation?

Forgetting to use the chain rule correctly.
Dropping constants or signs during algebraic simplification.
Not simplifying the final derivative before substituting values.
Forgetting to write full working — especially in exam questions worth multiple marks.

How do I find turning points?

To find turning points:

Differentiate the function to get dydx\frac{dy}{dx}dxdy.
Set dydx=0\frac{dy}{dx} = 0dxdy=0 and solve for xxx.
Use the second derivative test to determine if each point is a maximum, minimum, or point of inflection.

What are the main differentiation rules I need to know?

You must know the power rule, chain rule, product rule, quotient rule, and how to differentiate exponential and logarithmic functions. These form the foundation for all A-Level differentiation questions.