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8.03 Differentiation from first principles

Worksheet
Differentiation by first principles
1

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

a

Complete the limiting chord process for f \left( x \right) = x^{3} at the point x = 3 by filling in the table of values. Round your answers to four decimal places.

abh=b-a\dfrac{f\left(b\right)-f\left(a\right)}{b-a}
341
33.5
33.1
33.05
33.01
33.001
33.0001
b

Use the limiting chord process to calculate the instantaneous rate of change for the remaining values of x in the table:

x12345
\text{Instantaneous rate of change} f(x) \text{ at } x31248
c

Find the instantaneous rate of change of f \left( x \right) at any point x.

2

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

a

Complete the limiting chord process for f \left( x \right) = x^{2} at the point x = 3 by filling in the table of values:

abh=b-a\dfrac{f\left(b\right)-f\left(a\right)}{b-a}
341
33.5
33.1
33.05
33.01
33.001
33.0001
b

Use the limiting chord process to calculate the instantaneous rate of change for the remaining values of x in the table:

x12345
\text{Instantaneous rate of change} f(x) \text{ at } x248
c

Find the instantaneous rate of change of f \left( x \right) at any point x.

3

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

a

Complete the limiting chord process for f \left( x \right) = 2 x^{2} at the point x = 3 by filling in the table of values. Round your answers to four decimal places.

abh=b-a\dfrac{f\left(b\right)-f\left(a\right)}{b-a}
341
33.5
33.1
33.05
33.01
33.001
33.0001
b

Use the limiting chord process to calculate the instantaneous rate of change for the remaining values of x in the table:

x12345
\text{Instantaneous rate of change of }f(x) \text{ at } x4816
c

Hence, find the instantaneous rate of change of f \left( x \right) for any given value of x.

4

For each of the following functions:

i

Find f \left( x + h \right).

ii

Find f \left( x + h \right) - f \left( x \right).

iii

Find \dfrac{f \left( x + h \right) - f \left( x \right)}{h}.

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

For each of the following functions:

i

Find f \left( x + h \right) in expanded form.

ii

Find f \left( x + h \right) - f \left( x \right).

iii

Find \dfrac{f \left( x + h \right) - f \left( x \right)}{h}.

iv

Hence, find f' \left( x \right) by evaluating \lim_{h \to 0}\left(\dfrac{f \left( x + h \right) - f \left( x \right)}{h}\right).

a
f \left( x \right) = x^{2}
b
f \left( x \right) = 4 x^{2}
c
f \left( x \right) = - 6 x^{2}
6

Find the derivative of each of the following functions from first principles:

a
f \left( x \right) = 6
b
f \left( x \right) = - 6
c
f \left( x \right) = 5 x
d
f \left( x \right) = - 2 x
e
f \left( x \right) = 2 x + 3
f
f \left( x \right) = x^{2} - 3
g
f \left( x \right) = 4 x^{2} + 5
h
f \left( x \right) = x^{2} - 3 x
i
f \left( x \right) = 2 x^{2} - 3 x
j
f \left( x \right) = 4 x^{2} - 3 x - 5
k
f \left( x \right) = \sqrt{x}
l
f(x) = \sqrt{x - 5}
Gradients
7

Consider the function f \left( x \right) = 4 x.

a

Is the gradient of the function constant or variable?

b

Find the gradient of the function using the points \left(1, 4\right) and \left(7, 28\right) that lie on the function.

c

Find the gradient of the function from first principles.

8

Consider the function f \left( x \right) = \left(x + 6\right) \left(x + 2\right).

a

Find the gradient function of f \left( x \right) = \left(x + 6\right) \left(x + 2\right) from first principles.

b

Find the gradient at the point where x = 3.

9

Consider the function y = \sqrt{x}.

a

Find the gradient function of f \left( x \right) = \sqrt{x} from first principles.

b

Find the gradient at the point where x = 25.

10

Consider the function f \left( x \right) = 7 - x^{2}.

a

Find the gradient function f' \left( x \right) using first principles.

b

Hence, calculate the gradient of f \left( x \right) at the point where x = -1.

11

Consider the function f \left( x \right) = x^{2} + 3 x + 4.

a

Find the gradient function f' \left( x \right) using first principles.

b

Hence, calculate the gradient of f \left( x \right) at the point where x = 4.

12

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

a

Find the gradient function f' \left( x \right) using first principles.

b

Hence, calculate the gradient of f \left( x \right) at the point where x = 2.

13

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

a

Find the gradient function f' \left( x \right) using first principles.

b

Hence, calculate the gradient of f \left( x \right) at the point where x = - 5.

14

Find the gradient of f \left( x \right) = 3 x^{2} at the point \left(4, 48\right) from first principles.

15

Find the gradient of f \left( x \right) = 3 x^{3} at the point \left( - 4 , - 192 \right) from first principles.

16

Find the gradient of f \left( x \right) = 2 x \left(x - 6\right) at the point \left(2, - 16 \right) from first principles.

17

Find the gradient of f \left( x \right) = 3 x \left(x - 2\right)^{2} at the point \left(4, 48\right) from first principles.

18

Consider the function y = \dfrac{1}{x}. Find the x-values of the two points on the function at which the gradient is - \dfrac{1}{16}.

19

Consider the function f \left( x \right) = \dfrac{1}{2 x - 4}. Find f' \left( 4 \right).

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Outcomes

2.4.1.1

explore average and instantaneous rate of change in a variety of practical contexts

2.4.1.2

use a numerical technique to estimate a limit or an average rate of change

2.4.1.3

examine the behaviour of the difference quotient [𝑓(𝑥+ℎ)−𝑓(𝑥)]/h ℎ as ℎ→0 as an informal introduction to the concept of a limit

2.4.1.4

differentiate simple power functions and polynomial functions from first principles

2.4.1.5

interpret the derivative as the instantaneous rate of change

2.4.1.6

interpret the derivative as the gradient of a tangent line of the graph of 𝑦=𝑓(𝑥)

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