Introduction to differentiation 321
1.0
0.5
0
1
2
3
(b)
(a)
y 5 In x
4
5
6 x
1.5
2
y
0
21
22
1
2
1
2
3
4
5
6 x
dy 5
dx
1
x
dy
dx
Figure 34.8
Problem 20. Differentiate the following with
respect to the variable (a) y = 3e 2x (b) f (t ) =
4
3e 5t
(a) If y = 3e 2x then
dy
dx
= (3)(2e 2x ) = 6e 2x
(b) If f (t ) =
4
3e 5t =
4
3
e −5t , then
f
(t) =
4
3
(−5e
−5t
) = −
20
3
e
−5t
= −
20
3e 5t
Problem 21. Differentiate y = 5 ln3x
If y = 5 ln3x, then
dy
dx
= (5)
1
x
=
5
x
Now try the following Practice Exercise
Practice Exercise 135 Differentiation of e ax
and ln ax (answers on page 354)
1. Differentiate with respect to x: (a) y = 5e 3x
(b) y =
2
7e 2x
2. Given f (θ) = 5 ln2θ − 4 ln3θ, determine
f (θ).
3. If f (t ) = 4 lnt + 2, evaluate f (t ) when
t = 0.25
4. Find the gradient of the curve
y = 2e x −
1
4
ln 2x at x =
1
2
correct to 2
decimal places.
5. Evaluate
dy
dx
when x = 1, given
y = 3e 4x −
5
2e 3x + 8 ln5x. Give the answer
correct to 3 significant figures.
34.8 Summary of standard
derivatives
The standard derivatives used in this chapter are summarized in Table 34.1 and are true for all real values
of x.
Table 34.1
y or f (x)
dy
dx
or f (x)
ax
n
anx
n−1
sin ax
a cos ax
cos ax
−a sin ax
e ax
ae ax
ln ax
1
x
Problem 22. Find the gradient of the curve
y = 3x
2
− 7x + 2 at the point (1, −2)
If y = 3x 2 − 7x + 2, then gradient =
dy
dx
= 6x − 7
At the point (1, −2), x = 1,
hence gradient = 6(1) − 7 = −1
Problem 23. If y =
3
x 2 − 2 sin4x +
2
e x + ln 5x,
determine
dy
dx
y =
3
x 2 − 2 sin4x +
2
e x + ln 5x
= 3x
−2
− 2 sin 4x + 2e
−x
+ ln 5x