```Introduction to differentiation
mc-bus-introtodiff-2009-1
Introduction
This leaflet provides a rough and ready introduction to differentiation. This is a technique used to
calculate the gradient, or slope, of a graph at different points.
Given a function, for example, y = x2 , it is possible to derive a formula for the gradient of its graph.
We can think of this formula as the gradient function, precisely because it tells us the gradient of
the graph. For example,
when y = x2
2x
So, the gradient of the graph of y = x2 at any point is twice the x value there. To understand how
this formula is actually found you would need to refer to a textbook on calculus. The important
point is that using this formula we can calculate the gradient of y = x2 at different points on the
graph. For example,
when x = 3, the gradient is 2 × 3 = 6.
when x = −2, the gradient is 2 × (−2) = −4.
How do we interpret these numbers ? A gradient of 6 means that values of y are increasing at the
rate of 6 units for every 1 unit increase in x. A gradient of −4 means that values of y are decreasing
at a rate of 4 units for every 1 unit increase in x.
Note that when x = 0, the gradient is 2 × 0 = 0.
Below is a graph of the function y = x2 . Study the graph and you will note that when x = 3
the graph has a positive gradient. When x = −2 the graph has a negative gradient. When x = 0
the gradient of the graph is zero. Note how these properties of the graph can be predicted from
knowledge of the gradient function, 2x.
y
15
10
When x = -2 the gradient is negative
and equal to - 4.
- 4 - 3 -2 -1
When x = 3 the gradient is positive
and equal to 6
5
0
When x = 0 the gradient is zero.
x
1 2 3 4
Example When y = x3 , its gradient function is 3x2 . Calculate the gradient of the graph of y = x3
when a) x = 2, b) x = −1, c) x = 0.
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Solution
a) when x = 2 the gradient function is 3(2)2 = 12.
b) when x = −1 the gradient function is 3(−1)2 = 3.
c) when x = 0 the gradient function is 3(0)2 = 0.
You will need to use a notation for the gradient function which is in widespread use.
If y is a function of x, that is y = f (x), we write its gradient function as
dy
.
dx
dy
, pronounced ‘dee y by dee x’, is not a fraction even though it might look like one! This notation
dx
dy
can be confusing. Think of
as the ‘symbol’ for the gradient function of y = f (x). The process
dx
dy
is called differentiation with respect to x.
of finding
dx
Example
For any value of n, the gradient function of xn is nxn−1 . We write:
if
y = xn ,
then
dy
= nxn−1
dx
You have seen specific cases of this result earlier on. For example, if y = x3 ,
dy
= 3x2 .
dx
More notation and terminology
dy
, are y ′ , pronounced ‘y dash’,
When y = f (x) alternative ways of writing the gradient function,
dx
df
or
, or f ′ , pronounced ‘f dash’. In practice you do not need to remember the formulas for the
dx
gradient functions of all the common functions. Engineers usually refer to a table known as a Table
of Derivatives. A derivative is another name for a gradient function. The derivative is also known
as the rate of change of a function.
Exercises
1. Given that when y = x2 ,
dy
dx
= 2x, find the gradient of y = x2 when x = 7.
2. Given that when y = xn ,
dy
dx
= nxn−1 , find the gradient of y = x4 when a) x = 2, b) x = −1.
3. Find the rate of change of y = x3 when a) x = −2,
4. Given that when y = 7x2 + 5x,
dy
dx
b) x = 6.
= 14x + 5, find the gradient of y = 7x2 + 5x when x = 2.
1. 14.
2. a) 32,
b) −4.
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3. a) 12,
b) 108.
2
4. 33.
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