The Chemistry Maths Book, Second Edition

104 Chapter 4Differentiation

Linear combination of functions

A linear combination of the functions u, v, and wof xhas the form

y 1 = 1 au(x) 1 + 1 bv(x) 1 + 1 cw(x) (4.12)

where a, b, and care constants. Such a function can be differentiated term by term; by

Rules 1 and 2 in Table 4.3,

(4.13)

EXAMPLE 4.11Differentiatey 1 = 12 x

3

1 + 13 e

x

1 − 11 ln 1 x.

By Equation (4.13),

0 Exercises 27, 28

The product rule

The function

y 1 = 1 (2x 1 + 13 x

2

)(5 1 + 17 x

3

)

can be differentiated by treating it as the producty 1 = 1 uvwhereu 1 = 1 (2x 1 + 13 x

2

)and

v 1 = 1 (5 1 + 17 x

3

). Then, by Rule 3 in Table 4.3,

= 1 (2x 1 + 13 x

2

)(21x

2

) 1 + 1 (5 1 + 17 x

3

)(2 1 + 16 x)

This may now be simplified. In this example it is equally simple to multiply out the

original product and differentiate term by term, but in many cases the brute force

approach is more difficult than use of the product rule.

=+( )()()( )2 3 57 57+ ++ + 2 3

233 2

xx

d

dx

xx

d

dx

xx

dy

dx

u

d

dx

du

dx

=+

v

v

=+− 63

1

2

2

xe

x

x

dy

dx

d

dx

x

d

dx

e

d

dx

xxe

xx

=+− =×+×−× 23

1

2

23 3

1

2

1

32

ln

xx

1

2

dy

dx

a

du

dx

b

d

dx

c

d

dx

=++

vw