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2.04 The quotient rule

Worksheet
Quotient rule
1

Consider the function y = \dfrac{3}{x}.

a

By first rewriting it in negative index form, differentiate y.

b

By using the substitutions u = 3 and v = x, differentiate y using the quotient rule.

c

Find the value of x for which the gradient is undefined.

2

Consider the function y = \dfrac{2 x - 5}{5 x - 2}.

a

Using the substitution u = 2 x - 5, find u'.

b

Using the substitution v = 5 x - 2, find v'.

c

Hence find y'.

d

Is it possible for the derivative of this function to be zero?

3

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

a

Using the substitution u = 5 x^{2}, find u'.

b

Using the substitution v = 2 x + 8, find v'.

c

Hence find y'.

4

Differentiate the following functions using the quotient rule:

a
f(x)=\dfrac{7x}{8x-1}
b
y=\dfrac{4x^2}{3x-7}
c
y = \dfrac{3 x}{5 x - 4}
d
y = \dfrac{3 x^{2} + 2}{5 x^{2} + 4}
e
f(x)=\dfrac{2x+1}{x^2 -6x}
f
y=\dfrac{(x-5)^3}{8x}
g
f(x)=\dfrac{\sqrt{x-10}}{4x^2-6}
h
y=\dfrac{(2x-3)^4}{\sqrt{2x+5}}
i
y = \sqrt{\dfrac{2 + 7 x}{2 - 7 x}}
j
f \left( t \right) = \dfrac{\left( 4 t^{2} + 3\right)^{3}}{\left(5 + 2 t\right)^{5}}
k
y = \dfrac{x^{2} + 3 x - 2}{x + 2}
Gradients and tangents
5

Consider the function y = \dfrac{6}{\sqrt{x}} - 5.

a

Find the gradient function using the quotient rule.

b

Find the gradient of the function at x = 25.

6

Find the gradient of the tangent to the curve y = \dfrac{9 x}{4 x + 1} at the point \left(1, \dfrac{9}{5}\right).

7

Find the values of x such that the gradient of the tangent to the curve y = \dfrac{6 x - 1}{3 x - 1} is - 3.

8

Find the equation of the tangent to y = \dfrac{x}{x + 4} at the point \left(8, \dfrac{2}{3}\right).

9

Find the equation of the tangent to y = \dfrac{x^{2} - 1}{x + 3} at the point where x = 4.

10

Find the value of f' \left( 0 \right) for f \left( x \right) = \dfrac{x}{\sqrt{16 - x^{2}}}.

11

Find the value of f' \left( 4 \right) for f \left( x \right) = \dfrac{4 x^{7}}{\left(x + 4\right)^{4}}.

12

Find f' \left( 1 \right) for f \left( x \right) = \dfrac{6}{3 + 3 x^{2}} using technology or otherwise.

13

Find the value of f' \left( 3 \right) for f \left( x \right) = \dfrac{5 x}{9 + x^{2}} using technology or otherwise.

Increasing and decreasing functions
14

Consider the function y = \dfrac{1}{9 + x^{2}}.

a

Differentiate y using the quotient rule.

b

For what values of x is the function decreasing?

15

Consider the function y = \dfrac{3 - 4 x}{3 x - 4}.

a

Differentiate y.

b

Is it possible for the derivative to be zero?

c

Solve for the value of x that will make the derivative undefined.

16

Consider the function y = \dfrac{x + 6}{x - 6}.

a

Differentiate y.

b

Is it possible for the derivative to be zero?

c

Solve for the value of x that will make the derivative undefined.

d

Is the function increasing or decreasing near this value of x?

17

Consider the function y = \dfrac{5 x}{3 x - 4}.

a

Differentiate y.

b

Is it possible for the derivative to be zero?

c

Solve for the value of x that will make the derivative undefined.

d

Is the function increasing or decreasing near this value of x?

18

Consider the function y = \dfrac{2 x + 3}{2 x - 3}.

a

Differentiate y.

b

Is it possible for the derivative to be zero?

c

Solve for the value of x that will make the derivative undefined.

d

Is the function increasing or decreasing near this value of x?

19

Consider the function f \left( n \right) = \dfrac{1}{n + 3} + \dfrac{1}{n - 3}.

a

Differentiate f \left( n \right).

b

Is the function increasing or decreasing over its domain?

c

Find f' \left( 4 \right).

20

Consider the function f \left( x \right) = \dfrac{1}{4 + \sqrt{x}} + \dfrac{1}{4 - \sqrt{x}}.

a

Find the gradient function.

b

Is the function increasing or decreasing over its domain?

Applications
21

Differentiate y = \dfrac{x^{2}}{x + 3} and find the value of a if y' = 0 at x = a.

22

Differentiate y = \dfrac{x^{2} + k}{x^{2} - k} and find the possible values of k given that y' = 1 at x = - 3.

23

Consider the function g \left( x \right) defined as g \left( x \right) = \dfrac{f \left( x \right)}{x^{3} + 3}, where f \left( x \right) is a function of x.

Given that f \left( 2 \right) = 2 and f' \left( 2 \right) = 6, determine the value of g' \left( 2 \right).

24

Consider the identity 1 + x + x^{2} + \text{. . .} + x^{n - 1} = \dfrac{x^{n} - 1}{x - 1}, where x \neq 1 and n is a positive integer.

a

Form an expression for the sum 1 + 2 x + 3 x^{2} + \text{. . .} + \left(n - 1\right) x^{n - 2}.

b

Hence, find the value of 1 + 2 \times 5 + 3 \times 5^{2} + \text{. . .} + 8 \times 5^{7}.

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Outcomes

3.2.4.1

select and apply the product rule, quotient rule and chain rule to differentiate functions; express derivatives in simplest and factorised form

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