What does differentiation actually mean?

When you differentiate an equation, you're finding the gradient of its graph.

For example, if you differentiate the equation y = xyou get a solution dy/dx = 2x

This means that if you drew a line at a tangent to the curve of x2 at any point, and found the equation of that line in the form y = mx + c (where m is the gradient of the line, and c is the intercept) then the m value of that line would be 2x (with the x value at that point).

This makes sense; when y = 0 , the gradient of the curve is 0, and as x increases, y increases by 2x for every 1 that x increases by. Looking at the graph of x2, we can see that y does get bigger and increases more rapidly as x gets bigger; the slope or "differential" of the curve gets steeper.

But why is this useful?

Because the differential tells us the rate of change of x with y. It tells us how much y is changing as x changes, so it helps us to understand the relationship between x and y.

For example, imagine you're running a chemical reaction, with product "B". You want to make as much "B" as possible from your input "A". You know the relationship between A and B is given by B = 3A- 12A . 

Then you can find the differential dB/dA = 6A - 12 , which is positive so long as A is bigger than 2. As the differential is positive, then we know B is increasing, and we can see its increasing faster than A, as for every unit A increases, B increases by 6A-12. 

Therefore, you know that you want to make B in big batches, as you get more B out for every unit of A you put in. You also know that you definitely don’t want to make B with less than 2 units of A.

While this is a simple example, differentiation can be used on more complex equations in maths, physics, biology and chemistry to solve all kinds of problems.

Anna C. GCSE Maths tutor, A Level Maths tutor, GCSE Physics tutor, A ...

3 years ago

Answered by Anna, an A Level Maths tutor with MyTutor

Still stuck? Get one-to-one help from a personally interviewed subject specialist


£26 /hr

Daniel K.

Degree: Mathematical and Theoretical Physics (Masters) - Oxford, Merton College University

Subjects offered:Maths, Science+ 5 more

Further Mathematics
-Personal Statements-

“Mathematics and Theoretical Physics, University of Oxford. I enjoy sharing my experience and enthusiasm in Maths with those who could do with some help”

Tom H. A Level Economics tutor, A Level Maths tutor, A Level Spanish ...
£30 /hr

Tom H.

Degree: PGCE Secondary Mathematics (Other) - Durham University

Subjects offered:Maths, French+ 2 more

-Personal Statements-

“Highyl reviewed tutor from Durham University, ready to help you improve your grades, all the way to A*.”

PremiumTimothy N. A Level Design & Technology tutor, GCSE Design & Technolog...
£36 /hr

Timothy N.

Degree: Architecture and Environmental Engineering (Masters) - Nottingham University

Subjects offered:Maths, Physics+ 2 more

Design & Technology
-Personal Statements-

“Hi there, I have a passion for helping students achieve, and believe that with my 200+ hours of experience, we will be able to surpass the grades you want!”

About the author

Anna C. GCSE Maths tutor, A Level Maths tutor, GCSE Physics tutor, A ...

Anna C.

Currently unavailable: for new students

Degree: BEng Engineering Design with Study in Industry (Bachelors) - Bristol University

Subjects offered:Maths


“Hi, I'm Anna. I'm a friendly 4th year engineer at Bristol University keen to help anyone struggling with their Maths and Physics, and to try and make it fun as well!”

You may also like...

Other A Level Maths questions

How do I show two lines are skew?

Find the equation of a straight line that passes through the coordinates (12,-10) and (5,4). Leaving your answer in the form y = mx + c

Find the integral I of e^(2x)*cos*(x), with respect to x

Find the gradient of y=6x^3+2x^2 at (1,1)

View A Level Maths tutors

We use cookies to improve your site experience. By continuing to use this website, we'll assume that you're OK with this. Dismiss