MYTUTOR SUBJECT ANSWERS

361 views

What exactly IS differentiation?

This is a common question as we are often taught to differentiate by simply being told how to do the calculation, but not what it is we're really doing.

The concept

Differentiation is finding the gradient of a line. For simple problems this is easy to see using the normal trignometry methods for finding a gradient: Draw a graph of y = x and you can see that the gradient of the line is 1. Integrate y = x and you find that dy/dx = 1, as you would hope!

However the trignometric methods will not work for a curve such as y = x2 as the gradient is different at every point, so we have to use the differentiation. In this case what the differentiation is really doing is finding the gradient of the line at any given point x. This can be understood through differentiating by first principles.

First Principles

To differentiate y = x2 from first principles we begin by finding the gradient between point (x, x2) and an arbitrary point (x+h, (x+h)2).
This gives us the following equation:


which simplifies to:

Now for the important bit. We found the gradient between x and another point a distance h from x, but we want the gradient at x, so we find the limit of the above equation when h tends to 0. You can see on the graph below how reducing the value of h to 0 will give us the gradient at point x:

In our case of y = x2, this gives us the following answer:

Which gives us exactly what we would get by the differentiation methods we know!  You can try this with almost any differential function and you'll find that it works, but don't try anything too complicated just yet, as sometimes it can be tricky to evaluate that limit!

John H. GCSE Computing tutor, IB Further Mathematics  tutor, A Level ...

9 months ago

Answered by John, an A Level Maths tutor with MyTutor


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

260 SUBJECT SPECIALISTS

£20 /hr

Kirill M.

Degree: Physics (Masters) - Oxford, St Catherine's College University

Subjects offered: Maths, Physics+ 2 more

Maths
Physics
Further Mathematics
.PAT.

“Me Hi, I'm Kirill, I'm a 3rd year physicist at Oxford. Whether you are looking for exam preparation or for a better understanding of the material that you are covering in your studies of Maths or Physics, I can help you reach your goa...”

£20 /hr

Marco-Iulian G.

Degree: Mathematics&Computer Science (Masters) - Bristol University

Subjects offered: Maths, Further Mathematics + 2 more

Maths
Further Mathematics
.STEP.
.MAT.

“I'm in my first year at University of Bristol, studying Mathematics and Computer Science MEng. From an early age I started to participate in lots of contests and maths olympiads, and the experience I achieved along the way enriched bo...”

£20 /hr

Jake A.

Degree: Engineering (Masters) - Warwick University

Subjects offered: Maths, Physics+ 2 more

Maths
Physics
Further Mathematics
Chemistry

“Who am I? Hey! I’m a personal tutor, general maths enthusiast and am presently studying Engineering at Warwick University. I currently tutor students for Maths / Physics A-level along with Maths, Physics and Chemistry GCSE. I have a g...”

About the author

John H.

Currently unavailable: for regular students

Degree: Ship Science (Masters) - Southampton University

Subjects offered: Maths, ICT+ 1 more

Maths
ICT
Computing

“Who am I?A Ship Science (Engineering!) student at the University of Southampton, I have always lovedscience, maths and computers. I enjoy sharing that love, so I will do my best to show you why I think these things are so exciting!...”

You may also like...

Other A Level Maths questions

Let R denote the region bounded by the curve y=x^3 and the lines x=0 and x=4. Find the volume generated when R is rotated 360 degrees about the x axis.

A curve has equation y = 6ln(x) + x^2 -8x + 3. Find the exact values of the stationary points.

Differentiation: How to use the chain rule

The first term of an infinite geometric series is 48. The ratio of the series is 0.6. (a) Find the third term of the series. (b) Find the sum to infinity. (c) The nth term of the series is u_n. Find the value of the sum from n=4 to infinity of u_n.

View A Level Maths tutors

Cookies:

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

mtw:mercury1:status:ok