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8.04 Differentiation of y=x^n

Worksheet
Gradient function graphs
1

Consider the graph of y = x:

a

Find the gradient of the line at x = 4.

b

Find the gradient at any value of x.

c

What can be said about the gradient of a linear function?

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2

Consider the graph of y = x^{2} and its gradient function y = 2x:

a

What can be said of the sign of the gradient function when y = x^{2} is increasing or decreasing?

b

For x > 0, is the gradient of the tangent positive or negative?

c

For x \geq 0, as the value of x increases how does the gradient of the tangent line change?

d

For x < 0, is the gradient of the tangent positive or negative?

e

For x < 0, as the value of x increases how does the gradient of the tangent line change?

f

For y = x^{2}, the gradient of the tangent line changes at a constant rate. What type of function is the derivative of \\ y = x^{2}?

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3

Consider the graph of y = x^{3} and its gradient function y = 3x^2:

a

For x > 0, is the gradient of the tangent positive or negative?

b

For x \geq 0, as the value of x increases how does the gradient of the tangent line change?

c

For x < 0, is the gradient of the tangent positive or negative?

d

For x < 0, as the value of x increases how does the gradient of the tangent line change?

e

For y = x^{3}, the gradient of the tangent line first decreases at a decreasing rate, then increases at an increasing rate. What type of function is the derivative of y = x^{3}?

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4

Consider the graph of y = x^{4} and its gradient function y = 4x^3:

a

For x > 0, is the gradient of the tangent positive or negative?

b

For x \geq 0, as the value of x increases how does the gradient of the tangent line change?

c

For x < 0, is the gradient of the tangent positive or negative?

d

For x < 0, as the value of x increases how does the gradient of the tangent line change?

e

For y = x^{4}, the gradient of the tangent line is increasing, first at a decreasing rate and then at an increasing rate. What type of function is the derivative of \\ y=x^4?

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5

Consider the functions f \left( x \right) = x^{5} and g \left( x \right) = x^{4}.

a

Sketch the graph of f \left( x \right) and its derivative.

b

Sketch the graph of g \left( x \right) and its derivative.

c

If the degree of a function is even, will the degree of its derivative function be odd or even?

d

If the degree of a function is odd, will the degree of its derivative function be odd or even?

Algebraic differentiation
6

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

a

Find the derivative of f \left( x \right) = x^{2} from first principles.

b

The derivative of y = x^{n} for other values of n can be found using first principles in a similar way. The results for n = 2, 3, 4 and 5 are summarised in the following table:

yx^2x^3x^4x^5
y'2x3x^24x^35x^4

By observing the pattern in the table, find the derivative of the function g \left( x \right) = x^{8}.

c

Write the general form of the derivative of y = x^{n} for any value of n.

7
a

For each of the following functions, state the degree of the gradient function:

i

y = x^{2}

ii

y = x^{3}

iii

y = x^{4}

b

Hence, state the degree of the derivative of a polynomial function of degree n.

8

Find the derivative of the following functions with respect to x:

a
y = x^{7}
b
y = x^{9}
c
y = x^{ - 7 }
d
y = x^{ - 1 }
e
y = x^{\frac{6}{5}}
f
y = x^{\frac{2}{3}}
g
y = - 6
h
y = x^{0.6}
9

Consider the function y = \sqrt{x}.

a

Rewrite the function in index form.

b

Find the derivative of y = \sqrt{x}, in surd form.

10

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

a

Rewrite the function in negative index form.

b

Find the derivative of y = \dfrac{1}{x^{2}}.

11

Consider the function y = \dfrac{1}{\sqrt[4]{x}}.

a

Rewrite the function in negative index form.

b

Find the derivative of y = \dfrac{1}{\sqrt[4]{x}}, in positive index form.

12

Differentiate y = t \sqrt{t}.

13

Differentiate y = x^{ - \frac{1}{5} } with respect to x. Express your answer in positive index form.

Applications
14

Find the gradient of f \left( x \right) = x^{4} at x = 2. Denote this gradient by f' \left( 2 \right).

15

Consider the function f \left( x \right) = x^{9}. Find the equation of the tangent to the function at x = 1.

16

Consider the function f \left( x \right) = \dfrac{1}{\sqrt{x}}. Find the equation of the tangent to the function at x = 1.

17

David draws the graphs of x^{2}, x^{3}, x^{4} and x^{5} and draws the tangents to each one at the point where x = 1.

a

David then notices where each of the tangents cut the y-axis and records this in the table below:

\text{Graph}y\text{-intercept of the tangent line}\text{Gradient of the tangent}
y=x^2(0, -1)
y=x^3\left(0, -2\right)
y=x^4\left(0, -3\right)
y=x^5\left(0, -4\right)

Complete the table by calculating the gradient of each of the tangents at \\ x = 1.

b

Following the pattern in the table, what would be the gradient of the tangent to the graph of y = x^{n} at the point where x = 1?

c

Could the equation of the derivative of y = x^{6} be y' = x^{5}? Explain your answer.

18

The graph of y = x^{5} is shown below labelled as A. Fiona then graphs the derivative of the function, labelling it as B. She then finds the derivative of graph B to get graph C, then differentiates again to get graph D and differentiates again to get graph E.

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State whether the following statements are true about this sequence of derivatives:

a

The derivative of a function is always positive when the function is negative, and negative when the function is positive.

b

Each graph is a function of the form a x^{n}.

c

For any value of x, the value of the derivative will always be greater than the value of the function.

d

The degree of the derivative is always different to the degree of the function.

19

Lucy used first principles to find the derivatives of y = x^{\frac{1}{2}}, y = x^{\frac{3}{2}} and y = x^{\frac{5}{2}}. Her results are shown in the following table:

a

Find the derivative of y = x^{\frac{4}{3}}.

b

Find the derivative of y = x^{\frac{3}{4}}.

yx^{\frac{1}{2}}x^{\frac{3}{2}}x^{\frac{5}{2}}
\dfrac {dy}{dx}\dfrac{1}{2}x^{-\frac{1}{2}}\dfrac{3}{2}x^{\frac{1}{2}}\dfrac{5}{2}x^{\frac{3}{2}}
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Outcomes

2.4.2.1

examine examples of variable rates of change of non-linear functions

2.4.2.2

establish the formula 𝑑/𝑑x (𝑥^𝑛)=𝑛𝑥^(𝑛−1) for positive integers

2.4.2.5

calculate derivatives of power and polynomial functions

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