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Exercise on Differentiation
Any decimal answers should be given to 2 decimal places
Q1)
Write down the derivatives of the following functions:
a)
y = x
2
dy / dx =
b)
y = 4 x
2
dy / dx =
c)
y = 3 x
2
+ 7 x + 16
dy / dx =
d)
y =
–
4 x
2
–
7 x + 3
dy / dx =
e)
y = 12 x
2
–
7 x
–
20
dy / dx =
f)
y = x
3
dy / dx =
g)
y =5 x
4
dy / dx =
h)
y = x
-7
dy / dx =
i)
y = -5 x
-7
dy / dx =
j)
y =5 x
2
–
9 x
-4
+ 6
dy / dx =
k)
y =13 x
4
+
9 x
-2
+ 6 x
dy / dx =
l)
y = 6 x
5
+
2 x
-6
–
13 x
dy / dx =
m)
y =
–
3 x
-4
–
12 x
-2
–
6 x
dy / dx =
n)
y =5 x
8
+
3 x
-7
+ 9 x
dy / dx =
o)
y = 5 x
2
–
9 x
-4
+ 6 x
6
+ 34 x
dy / dx =
p)
y = 4 x
-4
–
2 x
-5
+ 3 x
5
+ 3 x + 5
dy / dx =
q)
y = -8 x
3
–
8 x
-3
+ 8 x
8
+ 8 x +8
dy / dx =
r)
y = -2 x
-2
–
9 x
-2
+ 6 x
2
+ 2 x
dy / dx =
s)
y =
–
x
-1
+
x
1
+ 6 x
2
+ 34 x
dy / dx =
t)
y = 28 x
-2
–
9 x
2
+ 2 x
9
+ 9 x
dy / dx =
Q2)
Calculate the gradient of the following functions at the point given:
a)
gradient of
y = 2 x
2
+ 3 x + 1
at
x = -2
b)
gradient of
y = 2 x
2
+ 3 x + 134
at
x = -2
c)
gradient of
y =
–
4 x
2
+ 5 x + 12
at
x = 4
d)
gradient of
y =
–
16 x
2
–
12 x + 34
at
x = 8
e)
gradient of
y =
–
3 x
-3
–
12 x
-4
–
4 x
at
x = 1
f)
gradient of
y =
–
3 x
-3
–
12 x
-4
–
4 x
at
x = -1
g)
gradient of
y =
2
x
-5
+
2 x
5
–
5 x
at
x = 3
h)
gradient of
y =
2
x
-5
+
2 x
5
–
5 x
at
x = -2
i)
gradient of
y =
2
x
4
–
12 x
-4
–
4 x
at
x = -3
j)
gradient of
y =
2
x
10
+
10 x
2
at
x = 2
Q3)
Find the point(s) at which the gradient of the following functions is zero:
a)
The gradient of
y = 2 x
2
+ 3 x + 1
is zero at:
b)
The gradient of
y = 3 x
2
–
3 x + 10
is zero at:
c)
The gradient of
y = 3 x
2
–
18 x + 14
is zero at:
d)
The gradient of
y = 3 x
2
+
18 x + 14
is zero at:
e)
The gradient of
y = 8 x
2
+ 12 x
–
14
is zero at:
f)
The gradient of
is zero at:
and
g)
The gradient of
is zero at:
and
h)
The gradient of
is zero at:
and
i)
The gradient of
is zero at:
and
j)
The gradient of
is zero at:
and
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