Harder implicit differentiation

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Simon Thomas 6 years ago

Created this as a copy of Simon's copy of SFY0004 Implicit 1.

Name	Status	Author	Last Modified
SFY0004 Implicit 1	Ready to use	Bill Foster	01/06/2016 09:47
Implicit Differentiation	draft	Clare Lundon	16/01/2018 13:19
Implicit 1	draft	Violeta CIT	26/09/2016 15:15
Simon's copy of SFY0004 Implicit 1	draft	Simon Thomas	11/03/2019 14:41
Harder implicit differentiation	draft	Simon Thomas	11/03/2019 15:41
Implicit 1	draft	Xiaodan Leng	11/07/2019 06:39

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Question statement

Find $\frac{dy}{dx}$ in terms of $x$ and $y$ using implicit differentiation in the following:

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			Name	Type	Generated Value

c	integer	5
b	integer	-5
a	integer	4

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$\var{a}xe^{\var{b}y}=(x+y)^\var{c}$

$\frac{dy}{dx}=$ Unnamed gap

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$\var{a}xe^{\var{b}y}=(x+y)^\var{c}$

If we differentiate the left hand side with respect to x we obtain:

$\frac{d}{dx}(\var{a}xe^{\var{b}y}) = e^{\var{b}y}\frac{d}{dx}(\var{a}x)+\var{a}x \frac{d}{dx}(e^{\var{b}y})$ by the product rule

$=\var{a}e^{\var{b}y}+\var{a*b}xe^{\var{b}y}\frac{dy}{dx}$ by the chain rule

If we differentiate the right hand side with respect to x we obtain:

$\frac{d}{dx}(x+y)^\var{c} = \var{c}(x+y)^\var{c-1}\frac{d}{dx}(x+y)$ by the chain rule

$=\var{c}(x+y)^\var{c-1}(1+\frac{dy}{dx})$

So implicitly differentiating both sides of our original equation with respect to $x$ gives:

$\var{a}e^{\var{b}y}+\var{a*b}xe^{\var{b}y}\frac{dy}{dx}=\var{c}(x+y)^\var{c-1}(1+\frac{dy}{dx})$

Collecting $\frac{dy}{dx}$ terms we obtain:

$(\var{a*b}xe^{\var{b}y}-\var{c}(x+y)^\var{c-1})\frac{dy}{dx}=\var{c}(x+y)^\var{c-1}-\var{a}e^{\var{b}y}$

Hence

$\frac{dy}{dx}=\frac{\var{c}(x+y)^\var{c-1}-\var{a}e^{\var{b}y}}{\var{a*b}xe^{\var{b}y}-\var{c}(x+y)^\var{c-1}}$

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This question is used in the following exam:

Implicit differentiation by Simon Thomas in Calculus.

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Simon Thomas 6 years ago

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Harder implicit differentiation

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Simon Thomas 6 years ago

Error in variable testing condition

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Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`