Quotient rule - differentiate quotient of linear terms

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Xiaodan Leng 5 years, 9 months ago

Created this as a copy of Quotient rule - differentiate quotient of linear terms.

Name	Status	Author	Last Modified
Quotient rule - differentiate quotient of linear terms	Ready to use	Newcastle University Mathematics and Statistics	20/11/2019 14:50
Luis's copy of Quotient rule - differentiate quotient of linear terms	draft	Luis Hernandez	30/11/2018 17:40
Quotient rule - differentiate quotient of linear terms	draft	Xiaodan Leng	10/07/2019 22:43

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Question statement

Let

$\simplify[std]{f(x) = ({a} * x+{b})/({c}x+{d})}$

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	2
c	integer	5
b	integer	-6
d	integer	4
s2	integer	1
s1	integer	-1
det	integer	38
c1	integer	5

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Generated value: `integer`

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Parts

Part a) Gap-fill
1. Gap Gap 0. Mathematical expression
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2. Gap Gap 1. Mathematical expression
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Part b) Gap-fill
1. Gap Gap 0. Mathematical expression
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2. Gap Gap 1. Mathematical expression
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Find numbers $a$ and $b$ such that

$\simplify[std]{f(x) = a + b/({c}x+{d})}$
Enter a and b as integers or fractions, but not as decimals.

$a=\;$ Gap 0.

$b=\;$ Gap 1.

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Advice

We have $\displaystyle \simplify[std]{{a}x+{b}={a}/{c}*({c}x+{d})+{b}-{a}*{d}/{c}={a}/{c}*({c}x+{d})+{-det}/{c}}$
Hence

$\begin{eqnarray*} \simplify[std]{({a} * x+{b})/({c}x+{d})}&=&\simplify[std]{({a}/{c}*({c}x+{d})+{-det}/{c})/({c}x+{d})}\\ &=&\simplify[std]{{a}/{c}+({-det}/{c})/({c}x+{d})} \end{eqnarray*}$
Where we have divided out by

$\simplify[std]{{c}x+{d}}$ at the last step.

We have

$\frac{dg}{dx} = \simplify[std]{{-c}/({c}x+{d})^2}$
using standard rules of differentiation.
Since from a),

$f(x) = \simplify[std]{{a}/{c}+({-det}/{c})/({c}x+{d})}$
we see that

$\begin{eqnarray*}\frac{df}{dx} &=&\simplify[std,!unitPower,!unitDenominator,!zeroFactor,!zeroTerm,!zeroPower]{(-{c})*(({-det}/{c})/({c}x+{d})^2)}\\ &=&\simplify[std]{{det}/({c}x+{d})^2} \end{eqnarray*}$

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Quotient rule - differentiate quotient of linear terms

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Xiaodan Leng 5 years, 9 months ago

Error in variable testing condition

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Generated value: `integer`

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Gap-fill

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Error

Quotient rule - differentiate quotient of linear terms

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Xiaodan Leng 5 years, 9 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`