Product 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 product rule.

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	4
s1	integer	1
b	integer	3
m	integer	6
n	integer	6

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Description

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Generated value: `integer`

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

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1. Gap Mathematical expression
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Gap-fill

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$\simplify[dPoly]{f(x) = ({a} + {b} * x) ^ {m} * e ^ ({n} * x)}$

You are given that

$\simplify[dPoly]{Diff(f,x,1) = ({a} + {b} * x) ^ {m -1} * e ^ ({n} * x) * g(x)}$

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

$g(x)=\;$ Unnamed gap

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Advice

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

$\simplify{Diff(u * v,x,1) = u * Diff(v,x,1) + v * Diff(u,x,1)}$

For this example:

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

$\simplify{v = e ^ ({n} * x)} \Rightarrow \simplify{Diff(v,x,1) = {n} * e ^ ({n} * x)}$

Hence on substituting into the product rule above we get:

$\simplify[dPoly]{Diff(f,x,1) = {m * b} * ({a} + {b} * x) ^ {m -1} * e ^ ({n} * x) + {n} * ({a} + {b} * x) ^ {m} * e ^ ({n} * x) = ({a} + {b} * x) ^ {m -1} * ({m * b + n * a} + {n * b} * x) * e ^ ({n} * x)}$

The last step was to take out the common term $\simplify[dPoly]{({a} + {b} * x) ^ {m -1} * e ^ ({n} * x)}$ .

Hence

$\simplify[dPoly]{g(x) = {m * b + n * a} + {n * b} * x}$ .

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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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Gap-fill

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Product 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`