Mathematical Methods for Physics and Engineering : A Comprehensive Guide

2.1 DIFFERENTIATION

Find the derivative with respect toxoff(t)=2at,wherex=at^2.

We could of course substitute fortand then differentiatefas a function ofx, but in this
case it is quicker to use

df

dx

=

df

dt

dt

dx

=2a

1

2 at

=

1

t

,

where we have used the fact that

dt

dx

=

(

dx

dt

)− 1

.

2.1.4 Differentiation of quotients

Applying (2.6) for the derivative of a product to a functionf(x)=u(x)[1/v(x)],

we may obtain the derivative of the quotient of two factors. Thus

f′=

(u

v

)′

=u

(

1

v

)′

+u′

(

1

v

)

=u

(

−

v′

v^2

)

+

u′

v

,

where (2.12) has been used to evaluate (1/v)′. This can now be rearranged into

the more convenient and memorisable form

f′=

(u

v

)′

=

vu′−uv′

v^2

. (2.13)

This can be expressed in words asthe derivative of a quotient is equal to the bottom

times the derivative of the top minus the top times the derivative of the bottom, all

over the bottom squared.

Find the derivative with respect toxoff(x)=sinx/x.

Using (2.13) withu(x)=sinx,v(x)=xand henceu′(x)=cosx,v′(x) = 1, we find

f′(x)=

xcosx−sinx

x^2

=

cosx

x

−

sinx

x^2

.

2.1.5 Implicit differentiation

So far we have only differentiated functions written in the formy=f(x).

However, we may not always be presented with a relationship in this simple

form. As an example consider the relationx^3 − 3 xy+y^3 = 2. In this case it is

not possible to rearrange the equation to giveyas a function ofx. Nevertheless,

by differentiating term by term with respect tox(implicit differentiation), we can

find the derivative ofy.