3.6The Chain Rule

Nothing we have done so far could help us take the derivative of this: But let’s try a function we know using a different approach: We can FOIL and then use the power rule: or…

Perhaps if we looked at these like fractions… …even if they can’t always be treated like fractions Look familiar?

The Chain Rule Perhaps if we looked at these like fractions… …even if they can’t 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 won’t go into that here but we will look at the other version of it with which you are all more familiar…

Also known to some as the “Outside-Inside” rule You’ve seen the Chain Rule written this way:

y in terms of x y in terms of u u in terms of x

Now let’s try a couple: Consider y to be a composite function that can be broken up or… Outside-InsideA.K.A: Don’t forget the Baby!

But where is the baby here? Now let’s go back to the first problem: or…

And now a really good one: The outside-inside rule can continue on forever… Or we can say that “babies” can grow up to have more “babies” Derivative of a base e function is always itself first Derivative of the square in sin 2 3x Derivative of the sine function Derivative of 3x Ready for this answer?

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 he’s 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 fast Herman is eating his In N Out fries How many words he says for every fry he eats How fast Herman is talking words/min

…and the key to solving that problem was… Every derivative problem can be thought of as a chain- rule problem: derivative of outside function derivative of inside function The derivative of x is one. UNITS!