Understanding Engineering Mathematics

B.Ifx=t^2 +1, y=t−1, evaluate

d^2 y

dx^2

as a function oft.

8.2 Revision

8.2.1 Geometrical interpretation of differentiation

➤

228 243➤

IfAandBare two points on a curve, the straight lineABis called achordof the curve
(157

➤
). A line which touches the curve at a single point (i.e.A=B) is called atangent
to the curve at that point (157
➤
). Thenormalto a curve at a pointAis a line through
A, perpendicular to the tangent atA.Thegradientorslope of a curveat a point is the
gradient or slope of the tangent to the curve at that point.

A

B

Tangent

Chord

Normal

Figure 8.1

Solution to review question 8.1.1

The derivative,

dy

dx

, is the slope or gradient of the curvey=f(x)at the

pointx, which is defined as tanθwhereθis the angle made by the tangent

to the curve atxwith the positivex-axis.

8.2.2 Differentiation from first principles

➤

228 243➤

AsBgets closer toA, the gradient of the chordABbecomes closer to that of the tangent
to the curve atA– that is, closer to the gradient of the curve atA. We say that in the
limit, asBtends toA, the gradient of the chord tends to the gradient of the curve atA.
We write this as

lim

B→A

(gradient chordAB)=gradient of curve atA

where limB→A means take the limiting value asB tends toA.Wemettheideaofa
limit on page 126, in the context of letting a number become infinitely large. Here we
are considering the situation where something – the distance betweenAandB– becomes
‘infinitesimally small’, i.e. tends to zero.
Let the equation of the curve bey=f(x). The situation is illustrated in Figure 8.2.
Supposeδx(‘deltax’) denotes a small change inx(not‘δtimesx’), so thatxbecomes
x+δx. Then correspondinglyf(x)will change tof(x+δx), producing a small change
inytoy+δy. The change iny,δy, is then given by

δy=y+δy−y=f(x+δx)−f(x)