Leicester: Differentiation 4

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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 functions $f(x)$ with respect to $x$ .

Input all numbers as fractions or integers, not as decimals.

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			Name	Type	Generated Value

a	integer	-3
c	integer	7
b	integer	2
d	integer	9
f	integer	2
ee	integer	0
u	integer	1
t	list	[ 0, 1 ]
v	integer	1

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Generated value: `integer`

-3

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1. Gap Mathematical expression
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$\displaystyle f(x)=\simplify[all]{(x^{a}-{b})/(x^{-c}+{d})}$

$\displaystyle \frac{df}{dx}=\;$ Unnamed gap

Advice

c) Let $\displaystyle \simplify[std]{u={u}*e^({c}x)+{1-u}*ln({c}*x)}$

So we have: $\displaystyle \frac{du}{dx}=\simplify[std]{{u*c}*e^({c}x)+{1-u}/x}$ and $\displaystyle f(u)=\simplify[std]{{t[0]}*sin(u)+{t[1]}*cos(u)}\Rightarrow \frac{df}{du}= \simplify[std]{{t[0]}*cos(u)-{t[1]}*sin(u)}$ .

By the Chain Rule

$\begin{eqnarray*} \frac{df}{dx}&=&\frac{df}{du}\times\frac{du}{dx}\\&=&\simplify[std]{{t[0]}*cos(u)-{t[1]}*sin(u)}\times\simplify[std]{{u*c}*e^({c}x)+{1-u}/x}\\&=&\simplify[std]{{t[0]}*({u}*{c}e^({c}x)+{1-u}/x)*cos({u}*e^({c}x)+{1-u}*ln({c}*x))-{t[1]}*({u}*{c}e^({c}x)+{1-u}/x)*sin({u}*e^({c}x)+{1-u}*ln({c}*x))}\end{eqnarray*}$

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This question is used in the following exams:

Leicester: Differentiation by Bill Foster in Bill's workspace.
Bradford: Questions on Differentiation by Bill Foster in Bill's workspace.
Questions on Differentiation by Sarah Jowett in Sarah's workspace.
Maths Support: Questions on differentiation by Christian Lawson-Perfect in Maths Support Wiki.
Differentiation by Gareth Woods in Gareth's workspace.
Jessica's copy of Maths Support: Questions on differentiation by Jessica Ryan in Jessica's workspace.
Chain Rule by Bill Foster in Bill's workspace.
Differentiation by Bill Foster in Bill's workspace.
Questions on differentiation by Marlon Arcila in Derivadas.

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Taxonomy: mathcentre

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Bill Foster 55 years, 2 months ago

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Generated value: `integer`

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Leicester: Differentiation 4

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`