Chain Rule Examples Examples using the Chain Rule
Chain Rule Examples
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- I'm now going to do a bunch more examples
- using the chain rule.
- So let's see.
- Once again.
- If I had f of x is equal to, let's see, I don't like this
- tool that I'm using now, let's have one of these.
- f of x is equal to, say, x to the third plus 2x squared
- minus, let's say, minus x to the negative 2.
- We haven't put any negative exponents in yet, but I
- think you'll see that the same patterns apply.
- And all of that to, let's say, the minus seven.
- We want to figure out what f prime of x, what the
- derivative of f of x is.
- So this might seem very complicated and daunting to
- you, and obviously to take this entire polynomial to
- the negative seventh power would take you forever.
- But using the chain rule, we can do it quite quickly.
- So the first thing we want to do, is we want to take the
- derivative of the inner function, I guess
- you could call it.
- We want to take the derivative of this.
- And what's the derivative of x to the third plus 2x squared
- minus x to the negative 2?
- Well, we know how to do that.
- That was the first type of derivatives we
- learned how to do.
- It's 3x squared and 2 times 2, plus 4x to the first, or just
- 4x, and then here, with a negative exponent, we do
- the exact same thing.
- We say negative 2 times negative 1, right, there's a 1
- here, we don't write it down.
- So negative 2 times negative 1 is plus 2 x to the, and then we
- decrease the exponent by 1, so it's x to the
- negative 3, right?
- So we figured out what the derivative of the inside is,
- and then we just multiply that, that whole thing, times
- the derivative of kind of the entire expression.
- So then that'll be, we take the minus 7, let me
- do a different color.
- So this is the entire thing.
- So then we take minus 7, so it's times minus 7, this
- whole expression, I'm going to run out of space.
- x to the third plus 2x squared minus x to the minus 2.
- That's minus x to the minus 2.
- And all of that, we just decrease this exponent
- by 1, to the minus 8.
- So let me write it all down a little bit neater now.
- So we get f prime of x as the derivative of f of x is equal
- to 3x squared plus 4x plus 2x to the minus third power, I
- don't know why did that.
- That's minus 3.
- Times minus seven times x to the third plus 2x squared minus
- x to the minus two, all of that to the negative eight power.
- And we could simplify it little bit.
- Maybe we could just multiply this minus 7, times, we could
- distribute it across this expression.
- So we'd say, that equals minus 7, so this equals minus 21 x
- squared, minus 28 x minus 14x to the negative 3.
- All of that times x to the third plus 2 x squared minus x
- to the minus 2 to the minus 8.
- So there we did it.
- We took this, what I would say is a very complicated function,
- and using the chain rule and just the basic rules we had
- introduced a couple of presentations ago, we were able
- to find the derivative of it.
- And now, if we wanted to, for whatever application, we could
- find the slope of this function at any point x by just
- substituting that point into this equation, and we'll get
- the slope at that point.
- Let me do a slightly harder one, to show you that the chain
- rule, you can kind of go arbitrarily deep in
- the chain rule.
- So let's say I had, let me see if I can write it
- a little bit thinner.
- If I had f of x, I don't know if you can see that, I'm going
- to do it a little fatter.
- f of x is equal to, I want to make it a little bit more
- complicated this time.
- 3x to the minus 2 plus 5 x to the third minus 7x, all of that
- to the fifth, and then this whole expression to
- the third power.
- So I imagine you saying, Sal, you're starting to go nuts,
- this is going to take us forever.
- Well, I'll show you, using the chain rule, it will
- not take that long.
- So the way I think about it, so-- f prime of x,
- f prime of x equals.
- I start off kind with the innermost function.
- So let me see if I can use colors to make it
- a little bit simpler.
- Let's take the derivative of this innermost function first.
- Actually, let me give you the big picture.
- We want to find the derivative of the innermost function, and
- then a little bit bigger, and then a little bit
- more big than that.
- I know that's not precise mathematical terms, but
- you'll get the point when I show you this example.
- So first we'll do this inner function, this
- inner expression.
- And the derivative of that's pretty easy, right?
- It's 15x squared minus 7, right?
- that was pretty straightforward.
- And now we're going to want to multiply that times this
- entire derivative here.
- So let me circle that in a different-- so then
- we want to do this.
- We're going to multiply that times this entire derivative.
- Well, that's just times 5.
- And we just pretend like this is just an x here, right?
- Because the derivative of x to the fifth is 5x
- to the fourth, right?
- But instead of an x, we have this whole expression, 5x
- to the third minus 7x.
- So we'll write that.
- 5x to the third minus 7x.
- Now the exponent here goes down by one.
- So it's 5 times 5x to the third minus 7x, all that
- to the fourth power.
- So we figured out the derivative of this so far, and
- then we want to figure out the derivative of this, so we'll
- add it, right, because we're trying to figure the derivative
- of this entire expression.
- So this is an easy one.
- Let me draw that in a different color.
- So we want the derivative of this.
- So that's negative 2 times 3, so that's negative
- 6x to the minus three.
- So what have we done so far?
- We've so far figured out the derivative of this entire
- expression, right?
- The derivative of that entire expression using
- the chain rule is this.
- And now, we're almost done.
- We just have to multiply that.
- So I'm going to just, I've run out of space on that line, but
- let's just assume that the line continues.
- So that's times.
- And now we just take the derivative of kind of
- this whole big thing.
- And now it's going to be the derivative of, I'm going
- to use this brown color.
- So it's a whole big expression to the third power, right?
- So that becomes times 3 times the whole expression, right?
- That's 3 times, now I'm going to write the whole thing, 3x to
- the minus 2 plus 5 x the third minus 7x, that to fifth, and
- then you decrement this by 1, to the second power.
- That was an ultraconfusing example, and this is probably
- the hardest chain rule problem you'll see in a lot of
- the questions you'll have on your test.
- You see, it wasn't that difficult.
- We just kind of went to the smallest possible function, and
- actually the smallest possible function would have been one of
- these terms, but we just found the derivative of this, which
- was 15 x squared minus 7, and then we just used the principle
- that the derivative of kind of a function is just the
- derivative of each of its parts-- well, actually, the
- derivative of-- we figured out the derivative of this inner
- piece, which was 15x squared minus 7, and then we multiplied
- it times the derivative of this slightly larger piece, which is
- 5 times this entire expression to the fourth, then we added it
- to the derivative of 3x to the minus 2.
- And then that whole thing, and actually I should put a big
- parentheses around here, that whole thing, we multiply it
- times the derivative of this larger expression.
- I think I might have confused you, so I apologize if I have,
- and in the next presentation I'm going to just do a bunch
- more chain rule problems, and at some point, it should
- start to make sense to you.
- I think it's just a matter of seeing example, after
- example, after example.
- I'll see you into the next presentation, and I apologize
- if I have confused you.
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