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2.01 Differentiation and the power rule

Worksheet
Power rule
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:

a
y = x - 9
b
y = 2 x + 9
c
y = x^{2} - 8 x - 6
d
y = x^{5} - x^{4} + 3
e
y = 2 x^{3} - 3 x^{2} - 4 x + 13
f
y = \dfrac{1}{2} x^{5} + \dfrac{1}{5} x^{8}
g
y = - 3 x^{5} + 5 x^{4} - 5 x^{3} - 4 x^{2} + 2 x - 4
h
y = 3 x^{ - 6 } + \dfrac{x^{ - 3 }}{7}
i
y = x^{\frac{1}{2}} + 8 x^{\frac{3}{4}}
j
y = \dfrac{x^4}{2} + 6x^{-\frac{1}{2}}
3

Differentiate y = \dfrac{2}{\sqrt{x}}. Express your answer in surd form.

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 y = \dfrac{1}{4 x^{3}}. Express your answer in positive index form.

6

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}.

7

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

8

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).

9

Consider the function y = \dfrac{8 x^{2} + 6 x + 4}{\sqrt{x}}.

a

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

b

Hence, find the derivative of the function.

10

For each of the following:

i

Express the function in expanded form.

ii

Find the derivative of the function.

a
y = \left( 8 x - 4\right)^{2}
b
y = x \left( 3 x + 4\right) \left( 5 x + 6\right)
c

y = \left( 6 x + 5\right) \left(x + 3\right)

d

y = 2 x^{2} \left( 7 x + 2\right)

e

y = \left(x + 4\right)^{2}

f

y = \dfrac{4}{9} \left( - 4 x - 8\right)

11

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.

12

For each of the following:

i

Rewrite the function in expanded form.

ii

Hence, find the derivative.

a

y = \left(\dfrac{4}{x} + 2 \sqrt{x}\right) \left(5 + \dfrac{6}{\sqrt{x}}\right)

b

y = \left(\sqrt[5]{x} + 2 \sqrt{x}\right)^{2}

c

y = \left( 3 \sqrt{x} + \dfrac{2}{x}\right)^{2}

d

y = \left( 4 \sqrt{x} - \dfrac{1}{\sqrt{x}}\right) \left( 4 \sqrt{x^{3}} + \dfrac{1}{x}\right)

e

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

Equations of tangents
13

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
14

Each of the following graphs contain a curve, f \left( x \right), along with one of its tangents, g \left( x \right).

i

State the coordinates of the point at which g \left( x \right) is a tangent to the curve f \left( x \right).

ii

State the gradient of the tangent.

iii

Hence, determine the equation of the line y = g \left( x \right).

a
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
b
-3
-2
-1
1
2
3
x
-6
-5
-4
-3
-2
-1
1
2
3
y
c
1
2
3
4
5
x
-1
1
2
3
y
d
1
2
3
4
5
x
-3
-2
-1
1
2
y
e
-1
1
2
3
4
5
x
1
2
3
4
5
y
f
-5
-4
-3
-2
-1
1
2
3
4
5
x
-4
-2
2
4
y
15

Consider the curve f \left( x \right) drawn along with g \left( x \right), which is a tangent to the curve:

a

What are the coordinates of the point at which g \left( x \right) is a tangent to the curve f \left( x \right)?

b

What is the gradient of the tangent line?

c

Hence, determine the equation of the line y = g \left( x \right).

d

What is the x-coordinate of the point on the curve at which we could draw a tangent line that has the same gradient as g \left( x \right)?

-3
-2
-1
1
2
3
x
-5
-4
-3
-2
-1
1
2
3
4
5
y
16

Consider the tangent to the curve f \left( x \right) = x^{2} at the point \left( - 1 , 1\right).

a

Find the gradient of the function f \left( x \right) = x^{2} at this point.

b

Find the equation of the tangent to the curve f \left( x \right) = x^{2} at the point \left( - 1 , 1\right).

17

Consider the tangent to the curve f \left( x \right) = x^{3} at the point \left( - 1 , -1\right).

a

Find the gradient of the function f \left( x \right) = x^{3} at this point.

b

Hence find the equation of the tangent to the curve f \left( x \right) = x^{3} at the point \left( - 1 , -1\right).

18

Consider the tangent to the curve f \left( x \right) = - x^{2} at the point \left( - 3 , - 9 \right).

a

Find the gradient of the function f \left( x \right) = - x^{2} at this point.

b

Hence, find the equation of the tangent to the curve f \left( x \right) = - x^{2} at the point \left( - 3 , - 9 \right).

19

Consider the tangent to the curve f \left( x \right) = - x^{3} at the point \left(2, - 8 \right).

a

Find the gradient of the function f \left( x \right) = - x^{3} at this point.

b

Hence, find the equation of the tangent to the curve f \left( x \right) = - x^{3} at the point \left(2, - 8 \right).

20

Consider the tangent to the curve f \left( x \right) = 3 x^{3} at x = 2.

a

Describe what must be done to find the equation of the tangent to the curve f(x) at \\ x=2.

b

Hence, find the equation of the tangent to the curve f \left( x \right) = 3 x^{3} at x = 2.

21

Consider the tangent to the curve f \left( x \right) = 3 x^{3} at x = 2.

a

State what two pieces of information we need in order to find the equation of the tangent line at x = 2.

b

Hence, determine the equation of the tangent to the curve f \left( x \right) = 3 x^{3} at x = 2.

22

Consider the tangent to the curve f \left( x \right) = 6 \sqrt{x} at the point \left(4, 12\right).

a

Find the gradient of the function f \left( x \right) = 6 \sqrt{x} at this point.

b

Hence, find the equation of the tangent to the curve f \left( x \right) = 6 \sqrt{x} at the point \left(4, 12\right).

23

Consider the tangent to the curve f \left( x \right) = 5 \sqrt{x} at the point \left(\dfrac{1}{9}, \dfrac{5}{3}\right).

a

Find the gradient of the function f \left( x \right) = 5 \sqrt{x} at this point.

b

Hence, find the equation of the tangent to the curve f \left( x \right) = 5 \sqrt{x} at the point \left(\dfrac{1}{9}, \dfrac{5}{3}\right).

24

Consider the tangent to the curve f \left( x \right) = - 2 x^{2} + 8 x + 2 at x = 1.

a

Find the y-coordinate of the point of intersection between the tangent and the curve.

b

Hence determine the equation of the tangent to the curve f \left( x \right) = - 2 x^{2} + 8 x + 2 at \\ x = 1.

25

Find the equation of the tangent to the curve f \left( x \right) = \dfrac{2}{x^{3}} at x = - 2.

26

Find the equation of the tangent to the curve f \left( x \right) = 0.3 x^{3} - 5 x^{2} - x + 4 at x = 1.

27

Find the equation of the tangent to the curve f \left( x \right) = \dfrac{9 x + 4}{3 x} at x = - 1.

Gradients
28

By considering the graph of f \left( x \right) = 2 x, find f'\left( - 5 \right).

29

Find the gradient of f \left( x \right) = x^{5} - 3 x^{4} at the point \left(3, 0\right).

30

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

31

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.

32

Consider the parabola f \left( x \right) = x^{2} + 3 x - 10.

a

Find the x-intercepts.

b

Find the gradient of the tangent at the positive x-intercept.

33

Consider the curve given by the function f \left( x \right) = x^{3} + 5 x.

Find the gradient of the tangent at the point \left(2, 18\right).

34

Consider the function y = 4 x^{2} - 5 x + 2. Find the value of x at which the tangent to the parabola is parallel to the x-axis.

35

Consider the function y = x^{2} - 3 x + 4. Find the x-coordinate of the point on the curve where the tangent makes an angle of 45 \degree with the x-axis.

36

Consider the function f \left( x \right) = 5 x^{2} + \dfrac{4}{x} - 1. The tangent to the curve at the point \left(2, 21\right) makes an angle of \theta with the x-axis. Find \theta, correct to the nearest degree.

37

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

a

Find the x-coordinate of the point at which f \left( x \right) has a gradient of 13.

b

Hence, state the coordinates of the point on the curve where the gradient is 13.

38

Consider the function f \left( x \right) = x^{3} + 3 x^{2} - 19 x + 2.

a

Find the x-coordinates of the points at which f \left( x \right) has a gradient of 5.

b

Hence, state the coordinates of the points on the curve where the gradient is 5.

39

Consider the function f \left( x \right) = 2 x^{2} - 216 \sqrt{x}. Find the coordinates of the point on the curve where the gradient is 0.

40

Find the x-coordinate(s) of the point(s) at which f \left( x \right) = \left(x - 5\right) \left(x^{2} + 3\right) has a gradient of 0.

41

Consider the function f \left( x \right) = x^{3} + 6 x^{2} - 14 x - 2.

a

Find the x-coordinate(s) of the point(s) on the curve where the gradient is the same as that of g \left( x \right) = 22 x - 4.

b

Hence, state the coordinates of the points on the curve where the gradient is the same as that of g \left( x \right) = 22 x - 4.

42

At point M, the equation of the tangent to the curve y = x^{2} is given by y = 4 x - 4. Find the coordinates of M.

43

At point M, the equation of the tangent to the curve y = x^{3} is given by y = 12 x - 16. Find the coordinates of M.

44

Consider the function f \left( x \right) = \dfrac{4 x^{3}}{3} + \dfrac{5 x^{2}}{2} - 3 x + 7. Find the x-coordinates of the points on the curve whose tangent is parallel to the line y = 3 x + 7.

45

5 x + y + 2 = 0 is the tangent line to the curve y = x^{2} + b x + c at the point \left(9, - 47 \right).

a

Find the derivative \dfrac{d y}{d x} of y = x^{2} + b x + c.

b

State the gradient of the tangent to the curve at x = 9.

c

Solve for the value of b.

d

Solve for the value of c.

46

The curve f \left( x \right) = k \sqrt{x} - 5 x has a gradient of 0 at x = 16. Find the value of k.

47

In the following graph, the line y = \dfrac{1}{10} x + b is tangent to the graph of f \left( x \right) = 6 \sqrt{x} at \\ x = a.

Find the values of a and b.

x
y
48

The curve y = a x^{3} + b x^{2} + 2 x - 17 has a gradient of 58 at the point \left(2, 31\right). Find the values of a and b.

49

The graph of y = a x^{3} + b x^{2} + c x + d intersects the x-axis at \left(2, 0\right), where it has a gradient of 36. It also intersects the y-axis at y = - 28, where the tangent is parallel to the x-axis.

Find the values of a, b, c and d.

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