300 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:

$\frac{dy}{dx}&space;=&space;\frac{x^{2}-(x+h)^{2}}{x-(x+h))}$
which simplifies to:

$\frac{dy}{dx}&space;=&space;2x+h$

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:

$\inline&space;\frac{dy}{dx}&space;=&space;\lim_{h\rightarrow&space;0}(2x&space;+&space;h)=2x$

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!

8 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

#### 177 SUBJECT SPECIALISTS

£20 /hr

Ben A.

Degree: Medicine (Bachelors) - University College London University

Subjects offered: Maths, Physics+ 6 more

Maths
Physics
Chemistry
Biology
.UKCAT.
-Personal Statements-
-Medical School Preparation-

“Hello,I am a first year medical student at University College London, When I was in sixth form, I helped tutor the younger years and I always enjoyed it, so I decided to keep it on in my university years.My subjects are the three m...”

£20 /hr

Linden S.

Degree: Economics and Finance (Bachelors) - Exeter University

Subjects offered: Maths, Economics

Maths
Economics

“I am currently in my third year studying economics and finance at the  University of Exeter. As well as having a strong passion for economics, I have a great love for mathematics too. I have experience in teaching through coaching ten...”

£20 /hr

Georgiana P.

Degree: Mathematics (Bachelors) - Bristol University

Subjects offered: Maths

Maths

“I am a Maths student at the University of Bristol and hope to share my love of maths with you. I believe that Maths is learnt in 3 steps:    1) I will explain and we will discuss the main principles of what we are working on, assis...”

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

How do I simplify surds?

Write sqrt(50) in the form Asqrt(50) where A is an integer

How do you differentiate y=x^x?

Solve the differential equation: e^(2y) * (dy/dx) + tan(x) = 0, given that y = 0 when x = 0. Give your answer in the form y = f(x).