mc-bus-diffrules-2009-1
Introduction
There are two rules known as linearity rules which, when used with a Table of Derivatives, enable us to differentiate a wider range of functions. These rules are summarised here.
Some notation
Before we look at the rules, we need to be clear about the meaning of the notation d d x
.
d y
When we are given a function y ( x ) and are asked to find we are being instructed to carry out d x an operation on the function y ( x ) . The operation is that of differentiation. A notation for this operation is used widely: d d x stands for the operation: ‘differentiate with respect to x ’
For example, d d x
( x 3 ) = 3 x 2 and d d x
( e 3 x ) = 3 e 3 x .
Differentiation of a function multiplied by a constant
If k is a constant and f is a function of x , then d d x
( kf ) = k d f d x
This means that a constant factor can be brought outside the differentiation operation.
Example
Given that d d x
( x
3 ) = 3 x
2 , then it follows that d d x
(7 x
3 ) = 7
× d d x
( x
3 ) = 7
×
3 x
2 = 21 x
2
Given that d d x
( e
2 x ) = 2 e
2 x , then it follows that d d x
(8 e
2 x ) = 8
× d d x
( e
2 x ) = (8)(2 e
2 x ) = 16 e
2 x www.mathcentre.ac.uk
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Differentiation of the sum or difference of two functions
If f and g are functions of x , then d d x
( f + g ) = d f d x d g
+ d x d d x
( f
− g ) = d f d x
− d g d x
This means that to differentiate a sum of two functions, simply differentiate each separately and then add the results. Similarly, to differentiate the difference of two functions, differentiate each separately and then find the difference of the results.
Example d y
Find d x
Solution when y = x 2 + x .
We require d d x
( x 2 + x ) . The sum rule tells us to differentiate each term separately. Thus d d x
( x
2 + x ) = d d x
( x
2 ) + d d x
( x ) = 2 x + 1 d y
So = 2 x + 1 .
d x
Example d y
Find d x
Solution when y = e 2 x
− x 5 .
The difference rule tells us to differentiate each term separately.
d d x
(e 2 x
− x
5 ) = d d x
(e 2 x )
− d d x
( x
5 ) = 2e 2 x
−
5 x
4
So d y d x
= 2e 2 x
−
5 x
4
Exercises
In each case use a Table of Derivatives and the rules on this leaflet to find d y d x
.
1.
y = e 5 x + 3 x 4
2.
y = x 2
− e
− x
3.
y = 3 x 2 + 7 x + 2
4.
y = 5 .
5.
y = 8e
−
9 x .
Answers
1.
5e 5 x + 12 x 3 , 2.
2 x + e
− x , 3.
6 x + 7 , 4. 0, 5.
−
72e
−
9 x .
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2 c math centre 2009