Chapter 3 Section 6

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3.6
The Chain Rule
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Nothing we have done so far could help us take
the derivative of this
But lets try a function we know using a
different approach
We can FOIL and then use the power rule
or
3
Perhaps if we looked at these like fractions
even if they cant always be treated like
fractions
Look familiar?
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The Chain Rule
Perhaps if we looked at these like fractions
even if they cant always be treated like
fractions
In fact, they can except that this version of the
chain rule does not apply to functions in which
perhaps the change in u might actually be 0.
however, the Chain Rule has been proven for all
functions. We wont go into that here but we
will look at the other version of it with which
you are all more familiar
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Youve seen the Chain Rule written this way
Also known to some as the Outside-Inside rule
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y in terms of x
y in terms of u
u in terms of x
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Consider y to be a composite function that can be
broken up
Now lets try a couple
or
Outside-Inside
A.K.A Dont forget the Baby!
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Now lets go back to the first problem
But where is the baby here?
or
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And now a really good one
Ready for this answer?
Derivative of a base e function is always itself
first
Derivative of the sine function
Derivative of 3x
Derivative of the square in sin2 3x
The outside-inside rule can continue on forever
Or we can say that babies can grow up to have
more babies
10
Jeffrey and Evan calculate that Herman is eating
his In N Out fries at a rate of 20 fries/minute.
Meanwhile, Eugene notes that Herman is also
talking while hes eating and deduces that he is
talking at a rate of 11 words for every fry that
he eats.
Use the Chain Rule to calculate how fast Herman
is talking in words/minute.
How many words he says for every fry he eats
How fast Herman is eating his In N Out fries
How fast Herman is talking
words/min
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and the key to solving that problem was
UNITS!
Every derivative problem can be thought of as a
chain-rule problem
The derivative of x is one.
derivative of outside function
derivative of inside function