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A collection of exercises designed to reinforce understanding of elementary calculus concepts, specifically differentiation and optimization. It includes a variety of problems involving differentiating functions, finding maximum and minimum values, and applying calculus techniques to real-world scenarios. The exercises are suitable for students studying introductory calculus at the high school or university level.
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Differentiate the following
functions
n
2
1
y
n
Y
n
4
mar
1
y
1
a a
blog
n Ye
1
n 4 ex
x
3
didae ex ex
2n
n x
anti n n n x 2 1
p
uv
y
v
u
1 3m
3m 2x 2m 2m
2x
11
e 4 da ont 2 dance tu
liggy
validayeudidav
e 4 3 3 2 ex
ex 4
2
39 12 3ne
ex 4
2
ex 3ne
ext 4
2
4
ex didn 2 log
n 2 logn
dida en
ex
2
2e ealogne
e2x
ex 2ne exalogn
nean
ex 1 2x alogn
mean
1 2x nlogse
nex
y
5h
1,
11
n
x n
2
1333
2m
e 5
do e
la
c x 15
dx
25
dida at 0
uvw u'vw uv'w ovw
m 9x 4m 36 m 9h 2m 8x
4m 12m 184 36
on 9h 18
t I
lte en i ek ex
item
2
en en en te
Iten
2
Iten
2
14
2
Jn 2
12 312
2n
25h 2
2
isn't salt
1
21m 35m
21m 24 20
7 412
gy
1
2
1 logse
logse
logn
312
isn't
Find the second order Derivative of the following
functions
3 Sat 3
1
18m 5
son
gy
mean a near area can 2n an
aea 2n an ean 2 2am
Zane a x e 29 Lane
named 2ean a x ean
Findthe Maximum Minimum values of the following
functions
x 8m 12
3
10x
equating
a
0 4m ion 0
un n2 4
un 0 x2 4 0
n 0 n 2
12m 10
o 1210
Manima
f 2 1214 16 48 16 32
0 Minima
f
1214 16 48 16 32 Minima
f
o o 810 12 12
Maximum Value
Minimum value
2 16 814 12 4 Minimum value
m m on 8
2n n 6
equating
14
0 2m x 6 0
2n x 2 31 2
2n 3 0 n 2
x
3
n 2
9
un
f
7 0 Manima
Minima
i
1
1
2 61
3
Maximum Value
31 9
033
Minimum value
3
2
6m on
É
61m x 2
equating too
x in 2 11 2
x
1 0 at 2 0
x 1 a
f i 1870
2 ieeo.im
i f
3
31112 11 Minimum value
1
Maximum value
19
3m 4
equating 0 3m use 1
0
3m 3m x 1 0
n 1 1 n 1 0
f
0 Minima
18 Minimum value
91
20m 15m
Sn 20
3
15m
0
sx m 4k 3
0
sx 0 m un
0
n
m 3m n
x n 3 1 n 3 0
n 3 0 n 1 0
n
a
1
20m 60m 30K
f
iii iii iiiiii.is
Maximum value occurs at a I minimum value at a
m 3n
t 3n
3m on
31m 2 1
equating
31m 2n
x 1
n n 1 1 n 1
x
9
2
on
It has no maximum or minimum value
Previous
questions
solved
Differentiate
a
let
c
Tn 5 use C
using
product rulefor 3 terms
of
uvw u'vw uv'w uvw
u Tx 5 u 5
w ex w ex
1
use
1
c
s e c su m s
ce
j
s
Je 5
realoga
j
s
su
Joe 5
Je 5
Differentiate
9
log
a
1
a 2
loga
an
2
a
loga
1
a
loga
an
2
Differentiate
shaded ea e
x
11
2ha e o e 8m
2rad ed ex at 8m
log
n
