10. Differentiating products
The sum and product rules
In Core 1 the idea of a rule for differentiating a function that was the sum of functions was devised. (Candidates will have used this before and probably not thought of it as a special rule).
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This can be justified using the same idea as was used in the previous section in these notes (regarding the Chain rule).
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Increasing "x" will increase all of the different "u" values, which will therefore increase "y":
Hence:
Hence:
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Taking limits can now give a derivative:
An assumption is made (that the limit of the sum is the same as the sum of the limits):
Hence:
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Now consider a function that is the product of other functions:
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This function cannot be differentiated using a rule known from these notes (until soon after this text), and therefore a rule must be developed to deal with this type of function.
Consider the changes in "y", "x", and the values of "u", as before:
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Unparseable or potentially dangerous latex formula. Error 5: Image dimensions are out of bounds: 2682x21
y + \delta y = [u_{1}u_{2} ... u_{n - 1}u_{n}] + \\ & [u_{1}u_{2} ... u_{n - 1} (\delta u_{n}) + u_{1}u_{2} ... (\delta u_{n - 1}) u_{n} + ... + (\delta u_{1}) u_{2} ... u_{n - 1} u_{n}] + \\ & [ ( u_{1}u_{2} ... (\delta u_{n - 1}) (\delta u_{n}) + u_{1}u_{2} ... (\delta u_{n - 2}) u_{n - 1} (\delta u_{n}) + ... + (\delta u_{1}) u_{2} ... u_{n - 1} (\delta u_{n}) ) + \\ & ( u_{1}u_{2} ... (\delta u_{n - 2}) (\delta u_{n - 1}) u_{n} + u_{1}u_{2} ... (\delta u_{n - 3}) u_{n - 2} (\delta u_{n - 1}) u_{n} + ... + (\delta u_{1}) u_{2} ... (\delta u_{n - 1}) u_{n} ) + ... + ( (\delta u_{1}) (\delta u_{2}) u_{3} ... u_{n - 1} u_{n} ) ] + ... + [( \delta u_{1} )( \delta u_{2} ) ... ( \delta u_{n - 1} ) ( \delta u_{n} )]
Hence:
Unparseable or potentially dangerous latex formula. Error 5: Image dimensions are out of bounds: 2500x21
\delta y = [u_{1}u_{2} ... u_{n - 1} (\delta u_{n}) + u_{1}u_{2} ... (\delta u_{n - 1}) u_{n} + ... + (\delta u_{1}) u_{2} ... u_{n - 1} u_{n}] + \\ & [ ( u_{1}u_{2} ... (\delta u_{n - 1}) (\delta u_{n}) + u_{1}u_{2} ... (\delta u_{n - 2}) u_{n - 1} (\delta u_{n}) + ... + (\delta u_{1}) u_{2} ... u_{n - 1} (\delta u_{n}) ) + \\ & ( u_{1}u_{2} ... (\delta u_{n - 2}) (\delta u_{n - 1}) u_{n} + u_{1}u_{2} ... (\delta u_{n - 3}) u_{n - 2} (\delta u_{n - 1}) u_{n} + ... + (\delta u_{1}) u_{2} ... (\delta u_{n - 1}) u_{n} ) + ... + ( (\delta u_{1}) (\delta u_{2}) u_{3} ... u_{n - 1} u_{n} ) ] + ... + [( \delta u_{1} )( \delta u_{2} ) ... ( \delta u_{n - 1} ) ( \delta u_{n} )]
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The "u" components are functions of "x", hence "y" is also a function of "x" (a composite function whose components are the "u" functions).
Limitations can be taken to find the derivative:
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Therefore get an expression for 
:
Unparseable or potentially dangerous latex formula. Error 5: Image dimensions are out of bounds: 2459x25
\frac{ \delta y }{ \delta x } = [u_{1}u_{2} ... u_{n - 1} ( \frac{ \delta u_{n} }{ \delta x }) + u_{1}u_{2} ... ( \frac{ \delta u_{n - 1} }{ \delta x }) u_{n} + ... + ( \frac{\delta u_{1}}{ \delta x }) u_{2} ... u_{n - 1} u_{n}] + \\ & [ ( u_{1}u_{2} ... (\frac{\delta u_{n - 1}}{ \delta x }) (\delta u_{n}) + u_{1}u_{2} ... (\frac{\delta u_{n - 2}}{ \delta x }) u_{n - 1} (\delta u_{n}) + ... + (\delta u_{1}) u_{2} ... u_{n - 1} (\frac{\delta u_{n}}{ \delta x }) ) + \\ & ( u_{1}u_{2} ... (\frac{\delta u_{n - 2}}{ \delta x }) (\delta u_{n - 1}) u_{n} + u_{1}u_{2} ... (\frac{\delta u_{n - 3}}{ \delta x }) u_{n - 2} (\delta u_{n - 1}) u_{n} + ... + (\frac{\delta u_{1}}{ \delta x }) u_{2} ... (\delta u_{n - 1}) u_{n} ) + ... + ( (\frac{\delta u_{1}}{ \delta x }) (\delta u_{2}) u_{3} ... u_{n - 1} u_{n} ) ] + ... + [( \frac{\delta u_{1}}{ \delta x } )( \delta u_{2} ) ... ( \delta u_{n - 1} ) (\delta u_{n})]
Hence:
Unparseable or potentially dangerous latex formula. Error 5: Image dimensions are out of bounds: 2538x25
\frac{dy}{dx} = \lim _{ \delta x \to 0 } ( [u_{1}u_{2} ... u_{n - 1} ( \frac{ \delta u_{n} }{ \delta x }) + u_{1}u_{2} ... ( \frac{ \delta u_{n - 1} }{ \delta x }) u_{n} + ... + ( \frac{\delta u_{1}}{ \delta x }) u_{2} ... u_{n - 1} u_{n}] + \\ & [ ( u_{1}u_{2} ... (\frac{\delta u_{n - 1}}{ \delta x }) (\delta u_{n}) + u_{1}u_{2} ... (\frac{\delta u_{n - 2}}{ \delta x }) u_{n - 1} (\delta u_{n}) + ... + (\delta u_{1}) u_{2} ... u_{n - 1} (\frac{\delta u_{n}}{ \delta x }) ) + \\ & ( u_{1}u_{2} ... (\frac{\delta u_{n - 2}}{ \delta x }) (\delta u_{n - 1}) u_{n} + u_{1}u_{2} ... (\frac{\delta u_{n - 3}}{ \delta x }) u_{n - 2} (\delta u_{n - 1}) u_{n} + ... + (\frac{\delta u_{1}}{ \delta x }) u_{2} ... (\delta u_{n - 1}) u_{n} ) + ... + ( (\frac{\delta u_{1}}{ \delta x }) (\delta u_{2}) u_{3} ... u_{n - 1} u_{n} ) ] + ... + [( \frac{\delta u_{1}}{ \delta x } )( \delta u_{2} ) ... ( \delta u_{n - 1} ) (\delta u_{n})] )
This can be split up like the addition rule. A lot of the terms do not concern the limit, (in fact all terms that do not include the term in the limit are not concerned with it). Also consider that if the limit of a product is the product of the limits of the factors, and there are factors of the form: 
. These are 0 as the "u" function depends on "x", and therefore cannot be taken out of the expression.
Hence:
![\frac{dy}{dx} = [ u_{1}u_{2} ... u_{n - 1} ( \frac{du_{n}}{dx} ) ] + [ u_{1}u_{2} ... ( \frac{du_{n - 1}}{dx} ) u_{n} ] + ... + [ u_{1} ( \frac{du_{2}}{dx} ) ... u_{n - 1}u_{n} ] + [ ( \frac{du_{1}}{dx} ) u_{2} ... u_{n - 1} u_{n}]](http://www.thestudentroom.co.uk/latexrender/pictures/79/79c84337f34bf4fb94a1662c0d56ad4e.png "\frac{dy}{dx} = [ u_{1}u_{2} ... u_{n - 1} ( \frac{du_{n}}{dx} ) ] + [ u_{1}u_{2} ... ( \frac{du_{n - 1}}{dx} ) u_{n} ] + ... + [ u_{1} ( \frac{du_{2}}{dx} ) ... u_{n - 1}u_{n} ] + [ ( \frac{du_{1}}{dx} ) u_{2} ... u_{n - 1} u_{n}]")
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This rule will most likely be encountered in the form:
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Differentiating quotients
Functions in the for 
can be differentiated using the product rule, as 
, however it is useful to have a rule for these functions.
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(Note that there is a multiplication of 
as this is due to the chain rule being applied; other wise there would be 
).
Hence:
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