Understanding Engineering Mathematics

corresponding increase iny:

δy=f(x+δx)−f(x)

and neglect any powers ofδxgreater than one to get something like

δyf′(x)δx

wheref′(x)denotes just a function ofxalone. Then

δy

δx

f′(x)

and lettingδx→0 yields the derivative

dy

dx

=f′(x)

as suggested by the notation. The solution to the review question illustrates this.

Solution to review question 8.1.2

If

y=f(x)=x^2 ,then

y+δy=f(x+δx)=(x+δx)^2

=x^2 + 2 xδx+(δx)^2

Hence

δy=f(x+δx)−f(x)= 2 xδx+(δx)^2

and

δy

δx

=

f(x+δx)−f(x)

δx

= 2 x+δx

So, in the limit, asδx→ 0

dy

dx

=

df

dx

= lim

δx→ 0

f(x+δx)−f(x)

δx

= lim

δx→ 0

2 x+δx

= 2 x

8.2.3 Standard derivatives

➤

229 243➤

By applying arguments such as that in the previous section we can build up a list of
standard derivatives. These are basically derivatives of the ‘elementary’ functions such
as powers ofx, trigonometric functions, exponentials and logs. What you remember from
these depends on the requirements of your course or programme, but I would certainly
recommend that the minimal list given in Table 8.1 be not only remembered, but be second
nature. Note, in anticipation of Chapter 9, that when you learn the standardderivatives
you can also learn thestandard integralsby reading the table from right to left (see
Section 9.2.2).