Quotient rule

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Bill Foster

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Bill Foster 55 years, 2 months ago

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Question statement

Differentiate the following function $f(x)$ using the quotient rule or otherwise.

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			Name	Type	Generated Value

a	integer	7
c	integer	5
b	integer	4
d	integer	5
s1	integer	1
d1	integer	5

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"Unnamed part" → "Unnamed gap" - Correct answer

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

You are given that

$\simplify[std]{Diff(f,x,1) = g(x) / (2 * ({c} * x + {d}) ^ (3 / 2))}$

for a polynomial $g(x)$ . You have to find $g(x)$ .

Input all numbers as fractions or integers.

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$g(x)=\;$ Unnamed gap

Advice

The quotient rule says that if $u$ and $v$ are functions of $x$ then

$\simplify[std]{Diff(u/v,x,1)=(v * Diff(u,x,1) -(u * Diff(v,x,1))) / v ^ 2}$

For this example:

$\simplify[std]{u = {a} * x + {b}}\Rightarrow \simplify[std]{Diff(u,x,1) = {a}}$

$\simplify[std]{v = Sqrt({c} * x + {d})} \Rightarrow \simplify[std]{Diff(v,x,1) = {c} / (2 * Sqrt({c} * x + {d}))}$

Hence on substituting into the quotient rule above we get:

$\simplify[std]{Diff(f,x,1) = ({a} * Sqrt({c} * x + {d}) -(({a} * x + {b}) * Diff(v,x,1))) / ({c} * x + {d}) = ({a} * Sqrt({c} * x + {d}) -(({c} * ({a} * x + {b})) / (2 * Sqrt({c} * x + {d})))) / ({c} * x + {d}) = ({2 * a} * ({c} * x + {d}) -({c} * ({a} * x + {b}))) / (2 * ({c} * x + {d}) ^ (3 / 2)) = ({a * c} * x + {2 * a * d -(c * b)}) / (2 * ({c} * x + {d}) ^ (3 / 2))}$

Hence

$\simplify[std]{g(x) = {a * c} * x + {2 * a * d -(c * b)}}$ .

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Taxonomy: mathcentre

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Bill Foster 55 years, 2 months ago

Error in variable testing condition

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Generated value: `integer`

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Quotient rule

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Bill Foster 55 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`