Basic Engineering Mathematics, Fifth Edition

Introduction to differentiation 321

1.0

0.5

0123

(b)

(a)

y 5 In x

456 x

1.5

2

y

0

21

22

1

2

123456 x

dy

dx^5

1

x

dy

dx

Figure 34.8

Problem 20. Differentiate the following with

respect to the variable (a)y= 3 e^2 x(b)f(t)=

4

3 e^5 t

(a) Ify= 3 e^2 xthen

dy

dx

=( 3 )( 2 e^2 x)= 6 e^2 x

(b) Iff(t)=

4

3 e^5 t

=

4

3

e−^5 t,then

f′(t)=

4

3

(− 5 e−^5 t)=−

20

3

e−^5 t=−

20

3 e^5 t

Problem 21. Differentiatey=5ln3x

Ify=5ln3x,then

dy

dx

=( 5 )

(

1

x

)

=

5

x

Now try the following Practice Exercise

PracticeExercise 135 Differentiation ofeax

andlnax(answers on page 354)

1. Differentiate with respect tox:(a)y= 5 e^3 x
   (b)y=

2

7 e^2 x

2. Given f(θ )=5ln2θ−4ln3θ, determine
   f′(θ ).
   3. If f(t)=4lnt+2, evaluate f′(t) when
   t= 0. 25
   4. Find the gradient of the curve
   y= 2 ex−

1

4

ln2x at x=

1

2

correct to 2

decimal places.

5. Evaluate

dy

dx

whenx=1, given

y= 3 e^4 x−

5

2 e^3 x

+8ln5x. Give the answer

correct to 3 significant figures.

34.8 Summary of standard derivatives

The standard derivatives used in this chapter are sum-

marized in Table 34.1 and are true for all real values

ofx.

Table 34.1

yorf(x)

dy

dx

orf′(x)

axn anxn−^1

sinax acosax

cosax −asinax

eax aeax

lnax

1

x

Problem 22. Find the gradient of the curve

y= 3 x^2 − 7 x+2 at the point( 1 ,− 2 )

Ify= 3 x^2 − 7 x+2, then gradient=

dy

dx

= 6 x− 7

At the point( 1 ,− 2 ),x=1,

hencegradient= 6 ( 1 )− 7 =− 1

Problem 23. Ify=

3

x^2

−2sin4x+

2

ex

+ln5x,

determine

dy

dx

y=

3

x^2

−2sin4x+

2

ex

+ln5x

= 3 x−^2 −2sin4x+ 2 e−x+ln5x