Basic Engineering Mathematics, Fifth Edition

Introduction to differentiation 315

chordJK.BymovingJnearer and nearer to

K, determine the gradient of the tangent of the

curve atK.

34.4 Differentiation from first principles

In Figure 34.4, Aand Bare two points very close
together on a curve,δx(deltax)andδy(deltay)rep-
resenting small increments in thexandydirections,
respectively.

0

y

f(x)

f(x 1

 x)

y

x

x

A(x, y)

B(x 1

 x, y 1

 y)

Figure 34.4

Gradient of chord AB=

δy
δx
however, δy=f(x+δx)−f(x)

Hence,

δy

δx

=

f(x+δx)−f(x)

δx

Asδxapproaches zero,

δy

δx

approaches a limiting value

and the gradient of the chord approaches the gradient of
the tangent atA.
When determining the gradient of a tangent to a curve
thereare two notationsused. The gradient of the curve
atAin Figure 34.4 can either be written as

limit

δx→ 0

δy

δx

or limit

δx→ 0

{

f(x+δx)−f(x)

δx

}

InLeibniz notation,

dy

dx

=limit

δx→ 0

δy
δx
Infunctional notation,

f′(x)=limit

δx→ 0

{

f(x+δx)−f(x)

δx

}

dy

dx

is the same asf′(x)and is called thedifferential

coefficientor thederivative. The process of finding the

differential coefficient is calleddifferentiation.

Summarizing, the differential coefficient,

dy

dx

=f′(x)=limit

δx→ 0

δy

δx

=limit

δx→ 0

{

f(x+δx)−f(x)

δx

}

Problem 3. Differentiate from first principles

f(x)=x^2

To ‘differentiate from first principles’ means ‘to find

f′(x)’ using the expression

f′(x)=limit

δx→ 0

{

f(x+δx)−f(x)

δx

}

f(x)=x^2 and substituting (x+δx) for x gives

f(x+δx)=(x+δx)^2 =x^2 + 2 xδx+δx^2 , hence,

f′(x)=limit

δx→ 0

{

(x^2 + 2 xδx+δx^2 )−(x^2 )

δx

}

=limit

δx→ 0

{

2 xδx+δx^2

δx

}

=limit

δx→ 0

{ 2 x+δx}

Asδx→ 0 ,{ 2 x+δx}→{ 2 x+ 0 }.

Thus,f′(x)= 2 xi.e.the differential coefficient ofx^2

is 2x.

This means that the general equation for the gradient of

the curvef(x)=x^2 is 2x. If the gradient is required at,

say,x=3, then gradient= 2 ( 3 )= 6.

Differentiation from first principles can be a lengthy

process and we do not want to have to go through this

procedureeverytimewewant todifferentiateafunction.

In reality we do not have to because from the above

procedure has evolved a setgeneral rule,whichwe

consider in the following section.

34.5 Differentiation ofy=axnby the

general rule

From differentiation by first principles, a general rule

for differentiatingaxnemerges whereaandnare any

constants. This rule is

ify=axnthen

dy

dx

=anxn−^1

or iff(x)=axnthenf′(x)=anxn−^1