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Differentiating Differentiating “Combined” “Combined” FunctionsFunctionsDeriving the Product Rule for Deriving the Product Rule for Differentiation.Differentiation.
Algebraic CombinationsAlgebraic Combinations
We have seen that it is fairly easy to compute the We have seen that it is fairly easy to compute the derivative of a “simple” function using the derivative of a “simple” function using the definition of the derivative. definition of the derivative.
More complicated functions can be difficult or More complicated functions can be difficult or impossible to differentiate using this method. impossible to differentiate using this method.
If we know the derivatives of two fairly simple If we know the derivatives of two fairly simple functions, we candeduce the derivative of an functions, we candeduce the derivative of an algebraic combination (e.g. the sum, product, algebraic combination (e.g. the sum, product, quotient) of these functions without going back quotient) of these functions without going back to the difference quotient.to the difference quotient.
A Gambier ExampleA Gambier Example
LARGE RATS!!!!
A Gambier ExampleA Gambier Example
( ) ( ) ( )Lg t k R t H t
The rate at which the deer gobblegobble my hostas is proportional to the product of the number of deer and the number of hostas. So we have a gobble function:
What can we say about the rate of change of this function?
What if over a short period of time t, from t to t + t the deer population increases by a small amount by RL and the hosta population increases by H. How much does the gobble rate change between time t and time t + t ?
A Gambier ExampleA Gambier Example
( ) ( ) ( )Lg t k R t H t
( ) ( ) ( )L Lg t t k R t R H t H ( ) ( ) ( ) ( )L L L Lk R t H t k R t H k R H t k R H
( ) ( ) ( ) ( )L L Lg t t g t k R t H k R H t k R H
So we need to compute ( ) ( ) :g t t g t
A Gambier ExampleA Gambier Example
( ) ( ) ( )Lg t k R t H t
( )Lk R t H
( )Lk R H t
Lk R H
The change in g has three relevant pieces:
Old Rats eating poor baby hostas
“Cute” baby rats eating vulnerable old hostas
“Cute” baby rats eating poor baby hostas
A Gambier ExampleA Gambier Example
0 0 0
( ) ( )lim lim limL L L
t t t
k R t H k R H t k R H
t t t
0
( ) ( )Now we consider g (t)= lim :
t
g t t g t
t
0 0
( ) ( )( ) ( )lim lim L L L
t t
k R t H k R H t k R Hg t t g t
t t
0 0 0 0( ) lim ( ) lim lim limL
L Lt t t t
RH Hk R t k H t k R
t t t
0
SO, the rate of change of SO, the rate of change of gobble is given by . . .gobble is given by . . .
0
( ) ( )lim ( ) ( ) ( ) ( )L L
t
g t t g tk R t H t k H t R t
t
The rate at which the number of hostas is changing times the number of large rats.
The rate at which the number of large rats is changing times the number of hostas.
The Product Rule for The Product Rule for DerivativesDerivatives
So the rate at which a product changes is not merely the product of the changing rates.
The nature of the interaction between the functions, causes the overall rate of change to depend on the size of the quantities themselves.
In general, we have…In general, we have…
0
( ) ( ) ( ) ( )( ) ( ) lim
h
f x h g x h f x g xdf x g x
dx h
0
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )limh
f x h g x h f x g x h f x g x h f x g x
h
0 0 0
( ) ( ) ( ) ( )lim ( ) lim ( ) limh h h
f x h f x g x h g xg x h f x
h h
0
( ) ( ) ( ) ( ) ( ) ( )limh
g x h f x h f x f x g x h g x
h
( ) ( ) ( ) ( )f x g x f x g x
In the course of the calculation above, we said that
Is this actually true? Is it ALWAYS true?
Continuity Required!Continuity Required!
0lim ( ) ( )h
g x h g x
x x+h
0lim ( )h
g x h
( )g x
The Limit of the Product The Limit of the Product of Two Functionsof Two Functions
0
( ) ( ) ( ) ( )( ) ( ) lim
h
f x h g x h f x g xdf x g x
dx h
0
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )limh
f x h g x h f x g x h f x g x h f x g x
h
0 0 0
( ) ( ) ( ) ( )lim ( ) lim ( ) limh h h
f x h f x g x h g xg x h f x
h h
0
( ) ( ) ( ) ( ) ( ) ( )limh
g x h f x h f x f x g x h g x
h
( ) ( ) ( ) ( )f x g x f x g x Would be zero if g were continuous at
x=a. Is it?
The Limit of the Product The Limit of the Product of Two Functionsof Two Functions
0
( ) ( ) ( ) ( )( ) ( ) lim
h
f x h g x h f x g xdf x g x
dx h
0
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )limh
f x h g x h f x g x h f x g x h f x g x
h
0 0 0
( ) ( ) ( ) ( )lim ( ) lim ( ) limh h h
f x h f x g x h g xg x h f x
h h
0
( ) ( ) ( ) ( ) ( ) ( )limh
g x h f x h f x f x g x h g x
h
( ) ( ) ( ) ( )f x g x f x g x g is continuous at x=a because g is
differentiable at x=a.
