Understanding Engineering Mathematics

• parametric differentiation


• higher order derivatives

Motivation

You may need the material of this chapter for:

• modelling rates of change, such as velocity, acceleration, etc.


• ordinary differential equations (Chapter 15)


• approximating the values of functions


• finding the maximum and minimum values of functions


• curve sketching

8.1 Review

8.1.1 Geometrical interpretation of differentiation ➤230 243➤➤

Complete the following:

The derivative,

dy

dx

,isthe of the curvey=f(x)at the pointx,whichis

defined as θ whereθis the angle made by the to the curve atx
with thex-axis.

8.1.2 Differentiation from first principles ➤230 243➤➤

Complete the following description of the evaluation of the derivative ofx^2 from first
principles:
Ify=f(x)=x^2 ,then

y+δy=f(x+δx)= +(δx)^2

Hence
δy=f(x+δx)−f(x)=

and
δy
δx

=

f(x+δx)−f(x)

δx

= +δx

So, ‘in the limit’, asδx→ 0

dy

dx

=

df

dx

= lim

δx→ 0

−f(x)