Maths Quest Maths B Year 12 for Queensland Chapter 4 Derivatives of exponential and logarithmic functions
WorkSHEET 4.2
WorkSHEET 4.2
1
Derivatives of exponential and logarithmic functions
Name: ___________________________
1
Differentiate each of the following:
(a) 32 x
(b)
(a)
y  32 x
Use the chain rule.
Let u  2 x then y  3u
log e 2 x  1
dy dy du


dx du dx
 log e 3  3u  2
2 log e 3  3 2 x
(b)
y  log e 2 x  1
Use the chain rule.
Let u  2 x  1, then y  log e u
dy dy du


dx du dx
1
 2
u
2

2 x  1
2
Differentiate each of the following:
(a) x 2 e x
(b)
(a)
y  x 2e x
Use the product rule with
u  x 2 and v  e x
dy du
dv

v u
dx dx
dx
2 x log e 2 x
 2 xe x  x 2 e x

 e x x 2  2x
(b)

y  2 x log e 2 x
Use the product rule with
u  2 x and v  log e 2 x
dy du
dv

v u
dx dx
dx
 2 log e 2 x  2 x 
 2 log e 2 x  2
1
x
Maths Quest Maths B Year 12 for Queensland Chapter 4 Derivatives of exponential and logarithmic functions
WorkSHEET 4.2
3
Differentiate each of the following:
ex
(a)
ex 1
(b)
4
e2x
x 1
Calculate
dy
if y =
dx
3e
(b)
log e 3e x  1
(a)

ex
e  1
2
e 2 x 2 x  3
x  12
y  3e x  32
Use the chain rule.

3

x
(b)
(a)
x
(a)
2
Let u  3e x  3, then y  u 2 .
dy dy du


dx du dx

 2u  3e x


 6ex 3e x  3
(b)


y  log e 3e x  1
Use the chain rule.
Let u  3e x  1, then y  log e u.
dy dy du


dx du dx
1
  3e x
u

3e x
3e  1
x
2
Maths Quest Maths B Year 12 for Queensland Chapter 4 Derivatives of exponential and logarithmic functions
WorkSHEET 4.2
5
Find the derivatives of the following
expressions:
1
(a)
1  log e x
(b)
y
(a)
1
1  log e x 
Use the chain rule.
Let u  1  log e x, then y 
 
log e e 3 x
1
.
u
dy dy du


dx d u d x
1
1


x 1  log e x 2
y  log e e 3 x
(b)
There are a number of methods to find
the derivative .
The simplest is to note that log e e 3 x  3x.
Therefore
6
Find the value of f 3 if f x  xe2 x2 .
dy
 3.
dx
f  x   xe2 x 2
f   x   e 2 x 2  2 xe2 x 2
 1  2 x e 2 x 2
When x  3, f 3  7e 6  2
 7e 4
 382
7
If N  N 0 e  kt and N = 1000 when t = 2, and
N = 2000 when t = 5, find values, correct to
two decimal places, for N 0 and k.
N  N 0 e  kt
1
2
2  1
1000  N 0 e 2k
2000  N 0 e 5k
2  e 3 k
log e 2  3k
1
k   log e 2  0.23
3
1000  N 0
N0
1
 2 log e 2
3
e
1
2 log e 2
 1000e 3
 629.96
3
Maths Quest Maths B Year 12 for Queensland Chapter 4 Derivatives of exponential and logarithmic functions
WorkSHEET 4.2
8
dy
 2 y and y = 12 when x = 0, find an
dx
expression for y in terms of x.
If
dy
 2 y, then y  Ae 2 x .
dx
If y  12 when x  0, then :
If
12  Ae 0
A  12
Therefore, y  12e 2 x
9
dP
dP
 kP and P = 45 when h = 0 and P = 32
If
 kP, then p  Ae kh .
dh
dh
when h = 120, find an expression for P in terms When h  0, P  45.
of h.
45  Ae 0
If
a  45
P  45e kh
P  32 when h  120
32  45e k120
 32 
120k  log e  
 45 
k  log e
3245 
120
k  0.00284
P  45e 0.00284 h
10
The half-life of C 14, an isotope of Carbon, is
Radioactive decay is exponential decay:
5700 years. This means half of a sample of C 14 N  N e  kt
0
will decay every 5700 years. If a sample
originally contained 40 grams of C 14, find an
If the half-life of C 14 is 5700 years then
expression for N, the amount of C 14 remaining
N0
after t years.
 N 0 e k 5700
2
 5700k  log e 0.5
k  0.00012
N  40e 0.0012t
4