Understanding Engineering Mathematics

function of a function rule to change the differentiation to one with

respect tot, and write

d^2 y

dx^2

=

d

dx

(

dy

dx

)

=

d

dt

(

dy

dx

)

×

dt

dx

=

d

dt

(

dy

dx

)/

dx

dt

on using

dt

dx

= 1

/

dx

dt

So:

d^2 y

dx^2

=

d

dt

(

1

2 t

)/

2 t

−

1

2 t^2

/

2 t=−

1

4 t^3

8.3 Reinforcement

8.3.1 Geometrical interpretation of differentiation

➤➤

228 230

➤

A.Evaluate the slopes of the following curves at the points specified:

(i) y=x^3 −xx= 1 (ii) y=sinxx=π

(iii) y= 2 ex x=0(iv)y=

3

x

x= 1

B.Determine where the slope of the curvey= 2 x^3 + 3 x^2 − 12 x+6 is zero.

8.3.2 Differentiation from first principles

➤➤

228 230

➤

Differentiate from first principles:

(i) 3x (ii) x^2 + 2 x+ 1 (iii) x^3 (iv) cosx

8.3.3 Standard derivatives

➤➤

229 232

➤

A.Differentiate without reference to a standard derivatives table:

(i) ex (ii) cosx (iii) x^31 (iv) lnx

(v) sinx (vi) x

1

(^3) (vii) tanx (viii)^1
x^3