Chain rule
Even More Chain Rule Even more examples using the chain rule.
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- Now that you've seen some examples of the chain rule
- in use, I think the actual definition of the chain rule
- might be more digestible now.
- So let me give you the actual definition of the chain rule.
- Let's say I have a function f of x and it equals h of g of x.
- And you remember all this from composite functions.
- So the chain rule just says that the derivative of f of x
- or f prime of x is equal to the derivative of this inner
- function, g prime of x times the derivative of this h
- function, h prime of x.
- But it's not going to be just h prime of x.
- It's going to be h prime of g of x.
- So let's apply that to some examples like we were
- doing before, and I think it'll make some sense.
- So let's say we had f of x is equal to x squared plus 5x
- plus 3, all of this to the fifth power.
- So in this example, what's h of x, what's g of x, and
- you know what f of x is.
- Well let's say g of x would be this inner function.
- So we would say-- let me pick a different color-- g of x here
- is x squared plus 5x plus 3.
- It's the stuff, f g of x.
- Well h of g of x is this whole thing, so what would h of x be?
- This is h of g of x, but h of x would just be x
- to the fifth, right?
- Because this expression as you took this entire g of x and you
- put it in for x right here.
- I think that make sense if you take this entire expression and
- you substitute x here for this entire expression you
- get this expression.
- And this shows that this is equal to h of g of x.
- If you just take this blue part and substitute it for x, you
- get this entire expression.
- So the chain rule just tells us that the derivative of this,
- that f prime of x-- and I have a feeling I'm going to run out
- of space-- f prime of x-- well actually before I do anything,
- let's figure out the derivatives of g
- of x and h of x.
- g of x of g prime of x-- let me draw a little line here to
- divide it out, I know I'm running out of space.
- So g prime of x is equal to 2x plus 5.
- 2x plus 5, and then derivative 3 is just 0, right?
- And the derivative of h of x? h prime of x is equal
- to 5 x to the fourth.
- So the chain rule just says that the derivative of this
- entire composite function is just-- let me just
- write it down here.
- I'm doing this to optimally confuse you.
- The derivative of this entire function is just g prime of x.
- Well we figured out with g prime of x is here, it's 2x
- plus 5 times h times h prime of g of x.
- So what's h prime of g of x?
- Well h prime of x is 5x to the fourth, but we want
- h prime of g of x.
- So h prime of g of x would equal 5 times g
- of x to the fourth.
- And we know what g of x is, it's this whole thing.
- So it would be times 5, and this whole thing x squared
- plus 5x plus 3, all that to the fourth power.
- I think I have truly, truly confused you, so I'm going
- to try to do a couple of more examples.
- Clear this.
- OK.
- Let me write it up here again.
- So if we say that f of x is equal to h of g of x, then f
- prime of x is equal to g prime of x times h prime of g of x.
- So I'll do another example.
- Let's say that g of x is equal to x to the seventh minus
- 3x to the ninth is 3.
- And let's say that h of x is equal to-- let's do something
- reasonably straightforward.
- Let's say h of x is x to the minus 10.
- So what is f of x? f of x is just h of g of x, and this
- should be a bit of a reminder from composite functions.
- So let's see.
- h of g of x would just be-- you take g of x and you substitute
- it for x here, so it would just be this expression, x to the
- seventh minus 3x to the minus third, and then all of that
- to the minus 10th power.
- So this is our f of x.
- And this is of course equal to f of x, right, because f of
- x is equal to h of g of x.
- I know this very confusing, but bear with me.
- Maybe you have to watch the video twice and it'll
- start making more sense.
- Well we want to now figure out what f prime of x is.
- Well the chain rule tells us all it is, is we take the
- derivative of g of x, right?
- So the derivative of g of x is what?
- That's easy.
- Or hopefully it's easy by now.
- Derivative of g of x is 7x to the sixth, and minus 3 times
- minus 3 is plus 9x to the minus 4.
- I just took minus 3 and went down 1, so that's g prime of x.
- And then times h prime of g of x.
- Well what's h prime of x?
- That's easy.
- That's just minus 10 times x to the minus 11.
- But we want to do h prime of g of x.
- So instead of having an x here, we're going to substitute that
- x with the entire g of x expression.
- So this is just times 10 time something to the minus eleven,
- and that something is just g of x.
- x to the seventh, minus 3x access to the minus 3.
- And there's our answer. f prime of x is the derivative of kind
- of the inner function, g of x, times the derivative of the
- outer function, but instead of it just being applied to x it'd
- be applied to the entire g of x instead of an x being here.
- Maybe I've confused you more.
- Let me do one quick example just to show you that you
- don't have to kind of do this whole h of g of every time.
- So if I have f of x is equal to 5 times minus x to the eighth,
- plus x to the minus eighth, all of that over to
- the fifth power.
- If I want to figure out f prime of x I just take the derivative
- of this inner function I guess I could call it, so that's
- minus 8x to the seventh minus 8x-- because it's just take the
- negative 8-- to the minus ninth, times the derivative
- of this larger function.
- So 5 times 5 is 25 times something to the fourth.
- And that something is just going to be this expression
- minus x to the eighth plus x to the minus eighth.
- And we're done.
- You could simplify it.
- You could multiply this 25 out and do et cetera, et cetera.
- Hopefully this gives you more of an intuition of what the
- chain rule is all about, and I'm going to do a lot more
- examples in the next couple of presentations as well.
- See you soon.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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