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26. Calculus (Extended)
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iGCSE (2021 Edition)

26.03 Power rule (ax^n) (Extended)

Lesson
Worksheet
Practice
Worksheet
Rules for differentiation
1

Differentiate the following:

a
y = 3 x^{2}
b
y = 6 x^{4}
c
y = -4 x^{3}
d
y = 5 x^{5}
e
y = - \dfrac{x^{3}}{5}
f
y = \dfrac{3x^{6}}{2}
g
y = \dfrac{5x^{4}}{12}
2

Differentiate the following, expressing your answers in positive index form.

a
y = \dfrac{14}{x \sqrt{x}}
b
y = \dfrac{1}{4 x^{3}}
3

Differentiate the following:

a
y = x - 9
b
y = 2 x + 9
c
y = \dfrac{2 x}{9} + 7
d
y = x^{2} - x + 8
e
y = x^{2} - 8 x - 6
f
y = x^{4} + x^{5}
g
y = x^{5} - x^{4} + 3
h
y = \dfrac{1}{2} x^{5} + \dfrac{1}{5} x^{8}
i
y = \dfrac{x^{8}}{8} + \dfrac{x^{5}}{5} - 3 x
4

For each of the following functions:

i
Rewrite the function in expanded form.
ii
Differentiate the function.
a
y = \dfrac{4}{9} \left( - 4 x - 8\right)
b
y = \left( 6 x + 5\right) \left(x + 3\right)
c
y = 2 x^{2} \left( 7 x + 2\right)
d
y = x \left( 3 x + 4\right) \left( 5 x + 6\right)
e
y = \left(x + 4\right)^{2}
f
y = 5\left(x - 3\right)^{2}
g
y = \left( 8 x - 4\right)^{2}
5

Differentiate the following functions:

a

y = \left( 3 x + 2\right) \left( 7 x + 6\right)

b

y = \left(x + 8\right) \left(x - 7\right) + 5

Gradients
6

Find f' \left( 2 \right) if f' \left( x \right) = 4 x^{3} - 3 x^{2} + 4 x - 6.

7

Consider the graph of f \left( x \right) = - 6 shown:

Find f' \left( 4 \right).

-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
8

Consider the graph of f \left( x \right) = 2 x - 3:

Find f'\left( - 4 \right).

-8
-6
-4
-2
2
4
6
8
x
-12
-10
-8
-6
-4
-2
2
4
6
y
9

The tangent to the curve y = 3 + \dfrac{x}{x + 2} at the point \left(0, 3\right) has the equation \\ y = \dfrac{1}{2} x + 3:

Find f' \left( 0 \right).

-1
1
2
3
4
x
1
2
3
4
5
y
10

Consider the graph of the function \\ f \left( x \right) = x^{2}:

a

How many points on the graph of f \left( x \right) = x^{2} have a gradient of 2?

b

Find the x-coordinate of the point at which f \left( x \right) = x^{2} has a gradient of 2.

-2
-1
1
2
x
-1
1
2
3
4
y
11

Find the gradient of f \left( x \right) = x^{4} + 7 x at the point \left(2, 30\right):

12

Consider the function f \left( x \right) = 6 x^{2} + 5 x + 2.

a

Find f' \left( x \right).

b

Find f' \left( 2 \right).

c

Find the x-coordinate of the point at which f' \left( x \right) = 41.

13

Consider the function f \left( x \right) = x^{3} - 4 x.

a

Find f' \left( x \right).

b

Find f' \left( 4 \right).

c

Find f' \left( - 4 \right).

d

Find the x-coordinates of the points at which f' \left( x \right) = 71.

14

Consider the function y = 2 x^{2} - 8 x + 5.

a

Find \dfrac{dy}{dx}.

b

Hence, find the value of x at which the gradient is 0.

15

Find the x-coordinates of the points at which f \left( x \right) = - 3 x^{3} has a gradient of - 81.

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Outcomes

0580E2.13A

Understand the idea of a derived function. Use the derivatives of functions of the form ax^n, and simple sums of not more than three of these.

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