What is 'Chain Rule' and why is it useful?

The chain rule is most commonly seen in Leibniz's notation, which is as follows:


dz/dx = dz/dy * dy/dx


You can remember it intuitively by thinking of the two 'dy' terms cancelling to leave dz/dx.


So why use the chain rule?

You are used to differentiating equations in the form y = f(x), but say both sides of the equation where functions eg g(y) = f(x) and you had to differentiate the equation with respect to x. 

g is a function of y, not x, so you can't simply calculate dg(y)/dx like you can df(x)/dx. Using the chain rule we can express dg(y)/dx as dg(y)/dy * dy/dx. These two terms can be calculated (assuming y is a function of x). This is really what the chain rule is saying: that the derivative of a function composition can be expressed as a product of the respective derivatives.


Another example of when the chain rule might come in useful is in mechanics: Acceleration is defined as the derivative of velocity: dv/dt. Sometimes though it might be useful to integrate acceleration of a distance, x, rather than over time. To eliminate time from this expression we can use the chain rule by saying dv/dt = dv/dx * dx/dt. Then noting that dx/dt is in fact velocity (v = dx/dt) we can write that dv/dt = v * dv/dx thus making acceleration a function only of velocity and position.

Tully K. A Level Maths tutor, GCSE Maths tutor, Uni Admissions Test -...

2 years ago

Answered by Tully, an A Level Maths tutor with MyTutor

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


Tiarnan B. GCSE Chemistry tutor, GCSE Maths tutor, GCSE Geography tutor
View profile
£20 /hr

Tiarnan B.

Degree: MChem (Masters) - York University

Subjects offered: Maths, Geography+ 1 more


“About Me: I am a student of Chemistry at the University of York studying for a masters degree and am a self-confessed science nerd! My goal is to give you the help you need, build your confidence and push you to be the best you can. A...”

MyTutor guarantee

Aidan T. GCSE Maths tutor, A Level Maths tutor
View profile
£22 /hr

Aidan T.

Degree: Chemical Engineering (Masters) - Edinburgh University

Subjects offered: Maths


“Hi! I am currently studying Chemical Engineering at The University of Edinburgh. I tutor in Edinburgh 6-7 times per week currently, and joined MyTutor as a way of sumplimenting that with the convenience of being able to stay in my ...”

PremiumShannon G. GCSE Biology tutor, A Level Biology tutor, GCSE Chemistry ...
View profile
£24 /hr

Shannon G.

Degree: Biomedical Science (Bachelors) - Oxford, Oriel College University

Subjects offered: Maths, Science+ 2 more


“Hello, I'm Shannon. I am an Oxford biomedical student and a passionate tutor. I am keen to offer support to any student studying chemistry, biology or maths with a love of learning!”

About the author

Tully K. A Level Maths tutor, GCSE Maths tutor, Uni Admissions Test -...
View profile

Tully K.

Currently unavailable: no new students

Degree: Engineering Science (Masters) - Oxford, Keble College University

Subjects offered: Maths, .PAT.


“Enthusiastic Engineering undergraduate at Oxford University. I'm passionate about maths and science and like to take a logical approach to these subjects and to tutoring.”

MyTutor guarantee

You may also like...

Other A Level Maths questions

If n is an integer prove (n+3)^(2)-n^(2) is never even.

How do you integrate xcos(x)?

How do I simplify [(x^2 + 4x- 12) / (x^2 - 25) ] / [(x+6) / (x^2-5x)] ?

What is the quotient rule and how is it applied?

View A Level Maths tutors


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