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8. Differential Calculus
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Year 11

8.05 Rules for differentiation

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Worksheet
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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}
h
y = 8x^{0.5}
2

Differentiate the following, expressing your answers in negative index form:

a

y = 8 x^{\frac{7}{9}}

b
y = 3 x^{0.6}
c
y = \dfrac{- 2}{x^{5}}
d
y = 7 x^{ - 3 }
3

Differentiate the following, expressing your answers in surd form:

a
y = 8 \sqrt{x}
b
y = \dfrac{\sqrt{x}}{4}
c
y = 14 x^{2} \sqrt{x}
d
y = \dfrac{2}{\sqrt{x}}
4

Consider the function y = \dfrac{7}{x}.

a

Rewrite the function in negative index form.

b

Find the derivative, expressing your answer with a positive index.

5

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

a
y = \dfrac{14}{x \sqrt{x}}
b
y = \dfrac{1}{4 x^{3}}
6

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 = 2 x^{3} - 3 x^{2} - 4 x + 13
i
y = \dfrac{1}{2} x^{5} + \dfrac{1}{5} x^{8}
j
y = - 3 x^{5} + 5 x^{4} - 5 x^{3} - 4 x^{2} + 2 x - 4
k
y = \dfrac{x^{8}}{8} + \dfrac{x^{5}}{5} - 3 x
l
y = 3 x^{ - 6 } + \dfrac{x^{ - 3 }}{7}
m
y = \dfrac{24}{x^{5}} - \dfrac{30}{x^{4}}
n
y = x^{\frac{1}{2}} + 8 x^{\frac{3}{4}}
7

Differentiate y = 7 a x^{7} - 2 b x^{3}, where a and b are constants.

8

Differentiate the following. Write your answers with positive indices.

a
y = x^{3} + x^{ - 5 } - 9
b
y = \sqrt[3]{x} + \dfrac{1}{x^{7}} - 4
c
y = \dfrac{1}{\sqrt[4]{x}} - x^{7} + \pi
d
y = x^{3} \sqrt{x} + 3 x^{5}
9

Consider the function f \left( r \right) = \dfrac{2}{r} + \dfrac{r}{3}.

a

Rewrite the function so that each term is a power of r.

b

Find f' \left( r \right).

10

Differentiate y = \dfrac{2}{x^{a}} - \dfrac{3}{x^{b}}, where a and b are constants.

11

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}
h
y = \left( 2 x + \dfrac{3}{x}\right) \left( 6 \sqrt{x} + 5\right)
12

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

13

Consider the function f \left( x \right) = \left(\sqrt{x} + 10 x^{2}\right)^{2}.

a

Rewrite the function f \left( x \right) in expanded form, with all terms written as powers of x.

b

Hence, differentiate the function.

14

Consider the following functions:

i

Rewrite the function so that each term is a power of x.

ii

Differentiate the function.

a
f \left( x \right) = \dfrac{8 x + 3}{2 x}
b
y = \dfrac{8 x^{9} - 4 x^{8} + 6 x^{7} + 9}{2 x^{2}}
c
y = \dfrac{3x^{6} +9 x^{4} + \sqrt{ x} + 11}{x^{3}}
d
y = \dfrac{8 x^{2} + 6 x + 4}{\sqrt{x}}
15

Consider the function y = \dfrac{5 x \sqrt{x}}{4 x^{5}}.

a

Rewrite the function in simplified negative index form.

b

Find \dfrac{dy}{dx}.

16

Differentiate the following functions, expressing your answers with positive indices:

a
y = \dfrac{x^{9} + 1}{x^{5}}
b
y = \dfrac{x^{2} + x^{13}}{x^{7}}
c
y = \dfrac{x + 1}{\sqrt[7]{x}}
d
y = \dfrac{\sqrt{x^3} + 5x}{\sqrt[3]{x}}
Gradients
17

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

18

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
19

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
20

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
21

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
22

Find the gradient of the following functions at the given point:

a

f \left( x \right) = 16 x^{ - 3 } at the point \left(2, 0\right).

b

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

c

f \left( x \right) = \dfrac{6}{\sqrt{x}} at the point \left(25, \dfrac{6}{5}\right).

d

f \left( x \right) = x^{3} - 2 x^{4} + \sqrt{x} at the point \left(4, - 446 \right).

e

f \left( x \right) = \dfrac{11}{x} + \dfrac{10}{x^{2}} at the point \left(4, \dfrac{27}{8}\right).

23

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.

24

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.

25

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.

26

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

27

Find the x-coordinate of the point at which f \left( x \right) = \sqrt{x} has a gradient of 6.

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Outcomes

2.4.2.3

understand the concept of the derivative as a function

2.4.2.4

recognise and use properties of the derivative 𝑑/𝑑x [𝑓(𝑥)+𝑔(𝑥)]=𝑑/𝑑x 𝑓(𝑥) + 𝑑/𝑑x 𝑔(𝑥)

2.4.2.5

calculate derivatives of power and polynomial functions

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