Hi guys, I'm Nancy
and I'm going to show you the Chain Rule, for finding derivatives.
I know you might not be super pumped about learning the Chain Rule
but here's why you should care...
the Chain Rule saves you time
and time is the one thing you can never get back.
So let's take a look at the Chain Rule.
So what is the chain rule?
And how do you even know when you need the Chain Rule?
Those are excellent questions.
What is the Chain Rule?
It is the way to find the derivative when you have
a more complicated equation that is...
a function inside a larger, outer function
and it will make your life easier, I promise.
So let's look at this one.
You need the Chain Rule when you have...
a smaller x expression or function inside a larger function
outer function
and the inside is something other than just x.
So this one is ripe for using the Chain Rule.
It is begging you to use the Chain Rule.
So lemme show you how you do it.
So the name "Chain Rule" is a little cryptic as to what to do
but thankfully it has another name, the "Outside-Inside Rule"
Outside-Inside Rule...
that tells you exactly what to do.
To find the derivative, dy/dx, or y prime...
you take the derivative of just the outside function first
the outer function
and leave the inside part as is, alone, untouched...
and then after that, you multiply by...
the derivative of just the inside function.
So let's try it with this one.
So dy/dx for this function, or y prime, if you see that instead...
will be equal to the derivative of just the outer function first...
in this example, the outer function is this power of 7
so you wanna focus on just that outer function form
the power of 7, and take the derivative of that.
You can just ignore the inside part for now.
To take the derivative of a power you use the Power Rule
which hopefully you've seen before
but it means that you...
bring down the power out front as a coefficient
so we bring down the 7 out front...
and you also...
knock down the power by 1, you reduce the power by 1
so that power becomes 6 instead of 7...
and for the Chain Rule
for now, you leave the inside part alone, the way it is
so this 3x + 1, just for now, will stay the same.
Now, you are not done with the Chain Rule
because you have something more than just x inside.
If this were x here, you would be done
but because there's an inner part
you have to multiply by the derivative of the inside function as well.
It also has a derivative component.
So, you focus on just that inside expression...
3x + 1...
and you take the derivative of just 3x + 1
ignoring the outer power.
Hopefully you've seen derivatives like this before.
The derivative will just be 3
because for this term, 3x...
it's the Constant Multiple Rule
the derivative of that would just be 3, the number out in front of the x.
And then for the plus 1 term
the derivative is just 0.
That part goes away, because the derivative of any constant is always 0.
So that whole derivative is just 3...
and you're pretty much done with the Chain Rule.
You can clean this up a little bit and simplify...
and that is your derivative
dy/dx = 21(3x + 1)^6
I combined the 7 times 3 out front.
If you were confused about the Power Rule
you can look that up, that's a separate topic, another video.
Basically, you bring that power down out front, as the coefficient
and then reduce the power by 1.
So that's how you do the Chain Rule, in general.
Now the question is...how do you know when you need it?
So for that, let's play a little game of Chain Rule or no Chain Rule.
So how do you know when you need the Chain Rule?
For this one we just did, we definitely needed the Chain Rule.
What if instead you'd been given a function like this, 3x + 1...
would you need the Chain Rule?
No, because that 3x + 1 is not inside some outer function...
so you don't need the Chain Rule because it is not a composite function
or composition of functions.
So no...
What about x^7?
You do not need the Chain Rule because it's just x to the seventh power.
If you had something in parentheses, more than x
raised to the seventh power, you would
but as it stands, it would just be overkill.
You do not need the Chain Rule.
(3x + 1)^7
That should look very familiar.
We used the Chain Rule in that, because...
it's an inside function in a larger, outer function.
It's a composite function. Definitely.
The inside derivative ended up being 3, and we didn't wanna miss that part
by not using the Chain Rule.
What about (x + 1)^7?
This one's a little tricky, 'cause...
technically, you should use the Chain Rule.
It's a composite function...
but when you do it, you'll see that that inside derivative is just 1
and multiplying by 1 won't actually make a difference
so it won't matter.
There are cases like this
but the thing is, it can't hurt to use the Chain Rule.
It's like, it's better safe than sorry.
So it's a good policy to use the Chain Rule, even then.
I'm gonna say yes.
Yes
and then what about (x^2 + 1)^7?
Definitely use the Chain Rule.
You'll get some inside derivative that is 2x
so it's important not to miss that.
Really quickly
some of you might be wondering, in this one that we just did
why couldn't you just multiply this all out
distribute it, FOIL it, and then take the derivative normally
like you'd been doing before with the Power Rule?
You can do that.
Feel free to do that.
Do it at your own peril, at your own risk
because it sounds like it will take a lot of time
be very tedious, and be kind of painful.
That is why the Chain Rule is your friend
because it's saving you that time and that pain
and it makes taking the derivative of something like this much, much faster.
So let's take a look at another type.
So here's another one, more complicated example.
What if you need to find the derivative of something like this?
It's the same steps as before. Use the Chain Rule.
If we wanna find the derivative of h(x), or h prime of x
sometimes that's called dh/dx, if you see that
but the derivative of h(x), h prime of x, equals...
let's use the Chain Rule again...
First take the derivative of the outside function
this outer part, the power of 9 only
so you'll use the Power Rule, and...
bring down the power to the front...
leave the inside the same, just leave it the way it is for now...
and reduce the power by 1
so instead of 9 you have a power of 8.
Remember you are not done
also need to multiply by the derivative of what is inside
of just that.
You'll use the Power Rule for this.
The derivative of x^2 + 5x + 6 will be...
2x from this part
plus a 5 from this term
and nothing from the constant
that contributes nothing to the inside derivative.
So the derivative of this will be 2x + 5...
you are actually done.
You don't really need to expand that or anything.
That is the answer for h'(x), the derivative.
Now let's play a really fast lightning round of Chain Rule or no Chain Rule.
So really quickly
would you need the Chain Rule for something like this?
No, it's not a composite function.
x^9? No, not complicated enough
don't need the Chain Rule.
(x - 6)^9?
Technically, yes.
It may give you an inside derivative of 1 that won't end up mattering, but
I'm gonna say yes. It's a good policy.
This one? Definitely, it's the one we just did
begging you to use the Chain Rule, that composite function.
So lemme show you one last kind of Chain Rule problem
I promise.
OK, here's another example.
It's trig, don't get too excited.
By now, you're probably getting the hang of this
but just humor me. Let's do this one, to find dy/dx...
It's the same steps as before, use the Chain Rule.
Take the derivative of the outside function
the outside function here is sine
it's the sine of something
and the derivative of sine is cosine
so we write cosine...
keep the inside part the same for now...
and then we need the inside derivative
and just focus on the x expression that's inside the sine function.
The derivative of x^2 - 3x is just 2x - 3...
and that's it.
That's your answer, dy/dx = cos(x^2 - 3x) * (2x - 3).
OK, what if instead you had just sin(x)
would you need the Chain Rule?
No, there's no complicated inside function.
Sin(3x)?
Yes, because this 3x is the inside function
within the larger sine function.
And what about sin(x^2)?
Yes, again the x^2 is inside the sine function
so you'll need the Chain Rule.
So you might be wondering, why do we even need to do the inside part?
Why couldn't we just do the cos(x^2 - 3x)
and be done with it, and not have to do that inside part?
That would be too easy.
This might be a good time for me to show you
the official formula for the Chain Rule...
OK, here are the official Chain Rule formulas
I have two versions here to show you
it's kind of what you already know, what you've been doing
this is a composite function
one function inside a larger, outer function
and if you have to take the derivative of that
you will need an outside derivative times an inside derivative.
That over there is Leibniz notation, if you have to use that
just so you know
this first part, dy/du, corresponds to the outside derivative
and this second part, du/dx, corresponds to the inside derivative
if you have to use that, if you're stuck with Leibniz notation
anyway, this inside derivative...inside derivative is important
you can't leave it out
or you won't have the right value for the overall derivative
overall rate.
If you want the derivative with respect to x
you have to keep taking the derivative
until you truly have it with respect to x
otherwise you're missing that part of the rate.
Why is it called the Chain Rule?
Because you could have many functions nested inside each other
more than just two
this is just g inside f
but it could keep going, g inside f inside e inside d, etc.
and if you had something like that
your derivative would just be this chain
of smaller derivatives multiplied together, smaller derivatives linked together
and you would need all of them to get the right value for the overall rate
the overall derivative
which brings me to my last thought.
So I lied, one more example
you may need to use the Chain Rule more than once
repeat the Chain Rule, Double Chain Rule, or extreme Chain Rule
this is where it gets pretty trippy, lemme show you
if you need to take the derivative of this
so y prime...
start just like you've started the others
take the derivative of the outside function
the square, the power of 2...
that power reduces, so you have a 1 power, which you don't need to write
take the derivative of the inside function, the (1 + cos2x)
the derivative of 1 is nothing, it's 0
but the derivative of cosine is negative sine
so we have -sin(2x)...
you are actually not done
because when you took the derivative of the inside
you got something that has its own inner function
this is another function nested inside, 2x.
Because you got something that is not just sin(x) or even -sin(x)
you have a 2x inside, you have to keep going
and use the Chain Rule again...
and find the derivative of that inside 2x
the derivative of 2x is just 2, so you're multiplying by 2...
so that's the answer, you had to use the Chain Rule twice.
The Chain Rule can come up in many forms
you may have to use the Chain Rule inside a Quotient Rule
inside the Product Rule
you just have to watch out for it.
You could even use the Chain Rule three times
you know, conceivably, I guess.
Lemme show you a few more forms where you'll need the Chain Rule.
Here's some other kinds to watch out for, where you'll need the Chain Rule
e^(3x), you have a 3x inside an exponential, outer, larger function
you'll need the Chain Rule
logs, this 5x is inside the natural log, ln
so you'll need the Chain Rule.
x^2 + 1 is underneath the square root
the square root is your outside, outer function
so you'll need the Chain Rule in that case.
You could also see this in that form of a fraction power
so the outside function is the power to the 1/2
and the inside function is x^2 + 1.
You just wanna watch out for the Chain Rule
it's really easy to forget it sometimes when you need it
so just watch out for things that are
smaller x expressions within larger function forms.
So I hope this video helped you understand how to use the Chain Rule.
I know calculus is everyone's favorite..
It's OK, you don't have to like math
but you can like my video!
So if you did, please click 'Like' or subscribe.
