Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PRELIMINARY CALCULUS

Find from first principles the derivative with respect toxoff(x)=x^2.

Using the definition (2.1),

f′(x) = lim

∆x→ 0

f(x+∆x)−f(x)

∆x

= lim∆x→ 0

(x+∆x)^2 −x^2

∆x

= lim

∆x→ 0

2 x∆x+(∆x)^2

∆x

= lim

∆x→ 0

(2x+∆x).

As ∆xtends to zero, 2x+∆xtends towards 2x, hence

f′(x)=2x.

Derivatives of other functions can be obtained in the same way. The derivatives

of some simple functions are listed below (note thatais a constant):

d

dx

(xn)=nxn−^1 ,

d

dx

(eax)=aeax,

d

dx

(lnax)=

1

x

,

d

dx

(sinax)=acosax,

d

dx

(cosax)=−asinax,

d

dx

(secax)=asecaxtanax,

d

dx

(tanax)=asec^2 ax,

d

dx

(cosecax)=−acosecaxcotax,

d

dx

(cotax)=−acosec^2 ax,

d

dx

(

sin−^1

x

a

)

=

1

√

a^2 −x^2

,

d

dx

(

cos−^1

x

a

)

=

− 1

√

a^2 −x^2

,

d

dx

(

tan−^1

x

a

)

=

a

a^2 +x^2

.

Differentiation from first principles emphasises the definition of a derivative as

the gradient of a function. However, for most practical purposes, returning to the

definition (2.1) is time consuming and does not aid our understanding. Instead, as

mentioned above, we employ a number of techniques, which use the derivatives

listed above as ‘building blocks’, to evaluate the derivatives of more complicated

functions than hitherto encountered. Subsections 2.1.2–2.1.7 develop the methods

required.

2.1.2 Differentiation of products

As a first example of the differentiation of a more complicated function, we

consider finding the derivative of a functionf(x) that can be written as the

product of two other functions ofx, namelyf(x)=u(x)v(x). For example, if

f(x)=x^3 sinxthen we might takeu(x)=x^3 andv(x)=sinx. Clearly the