Understanding Engineering Mathematics

convenient to do this – with care. For example in differential equations we sometimes use
such steps as:

dy

dx

dx≡dy

which is really a bit of poetic license that is found to work most of the time.
We can combine the product and chain rules to obtain the rule for differentiating a
quotient as follows:

d

dx

(u

v

)

=

d

dx

(uv−^1 )=v−^1

du

dx

+u

d(v−^1 )

dx

by the product rule

=v−^1

du

dx

+u

(

−v−^2

dv

dx

)

by the chain rule

=

1

v

du

dx

−

u

v^2

dv

dx

=

(

v

du

dx

−u

dv

dx

)/

v^2

When first learning these rules you might use the given formulae, substitutingu=x,
v=sinx, or whatever. However, the rules will be more useful to you if you practice and
develop your skills to the point where you do not need to do this. That is, rather than
actually use the formula, try to do the required procedure automatically. To help with this
think of the rules in terms of words, describing what you actually do:

Sum rule: ‘diff(+)=diff+diff’

Product: ‘diff(×)=one×diff+diff×other’

Quotient diff(/)=

diff top×bottom - top×diff bottom

(bottom)^2

Function of a function ‘diff[f(g)]=(difff w.r.t.g)×(diffgw.r.t.x)’

Solution to review question 8.1.4

(i)y= 3 x^4 − 2 x^2 + 3 x− 1

This is a linear combination of terms of the formAxn. We differen-

tiate each term in the polynomial and combine the results.

So

dy

dx

= 3 × 4 x^3 − 2 × 2 x+ 3

= 12 x^3 − 4 x+ 3

(ii) y=x^2 cosx is a product of the two elementary functions x^2

and cosx.