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8.04 Differentiating x^n for integer n

Worksheet
Differentiation with integer powers
1

Consider the function f \left( x \right) = x^{n}, where n is any positive integer.

Complete the following table:

f(x)f'(x)
x
x^2
x^3
x^4
x^n
2

Find the derivative of y = - 6 with respect to x.

3

Find the derivative of the following:

a
y = x^{7}
b

y = x^{ - 7 }

c

y = x - 9

d

y = 7 x^{ - 3 }

e

y = 5 x^{5}

f

y = x^{2} - x + 8

g

y = x^{4} + x^{5}

h

y = - \dfrac{1}{5} x^{3}

i
y = 7 x^{2} - 9 x + 8
j

y = \dfrac{1}{2} x^{5} + \dfrac{1}{5} x^{8}

4

Consider the function y = \dfrac{1}{x^{2}}.

a

Rewrite the function in negative index form.

b

Hence, find the derivative of y = \dfrac{1}{x^{2}}.

5

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

a

Rewrite the function in negative index form.

b

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

6

Differentiate each of the following, giving your answer with a positive index:

a

y = x^{ - 5 } + 7

b

y = x^{ - 5 } + x^{7}

c
y = \dfrac{6}{x^{2}}
d

y = \dfrac{1}{4 x^{3}}

e

y = \dfrac{x^{9} + 1}{x^{5}}

7

Differentiate the following functions:

a

y = \dfrac{- 2}{x^{5}}

b

f \left( x \right) = 2 x + 9

c

y = 5 x^{4} + 3 x^{2} + 4 x

d

y = 2 x^{3} - 3 x^{2} - 4 x + 13

e

f \left( x \right) = \dfrac{x^{8}}{8} + \dfrac{x^{5}}{5} - 3 x

f

f\left( x \right) = 3 x^{ - 6 } + \dfrac{x^{ - 3 }}{7}

g

y = \dfrac{24}{x^{5}} - \dfrac{30}{x^{4}}

h

y = \dfrac{- 5 x + 1}{8}

i

y = \dfrac{5 x^{2} - 4 x + 1}{3}

j

f \left( x \right) = \dfrac{2 x}{9} + 7

8

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

9

Differentiate y = \dfrac{2}{x^{a}} - \dfrac{3}{x^{b}}, where a and b are constants, giving your answer with a positive index.

10

Find the derivative of each of the following functions:

a

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

b
f \left( x \right) = \dfrac{8 x + 3}{2 x}
c
y = \dfrac{8 x^{9} - 4 x^{8} + 6 x^{7} + 9}{2 x^{2}}
d
f \left( x \right) = \dfrac{12x^3 + 2x^2-6}{3x^3}
11

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)

12

Differentiate the function f \left( x \right) = \left(x + 2\right)^{3} by expanding it first.

Gradients
13

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

14

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

15

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

16

By considering the graph of f \left( x \right) = 2 x - 3, find f'\left( - 4 \right).

17

Find the gradients of the following functions at x=2:

a
f \left( x \right) = x^{4}
b
f \left( x \right) = 16 x^{ - 3 }
c
f \left( x \right) = 6 x^{3}
18

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

19

Consider 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(s) of the point(s) at whichf \left( x \right) = x^{2} has a gradient of 2.

20

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.

21

Find the x-coordinate of the point at which f \left( x \right) = 5 x^{2} has a gradient of 10.

22

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

a

Find \dfrac{dy}{dx}.

b

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

23

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.

24

Find the x-coordinate(s) of the point(s) at which f \left( x \right) = x^{4} has a gradient of 108.

25

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

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MA11-1

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