Hello, and welcome to MySecretMathTutor, in this video we are going to take on a very
difficult topic, we're going to working on understanding implicit differentiation
Now, if you seen this before , you know that
it can be a very tricky process, but I'll give you a really big hint on how this is
going to play out, its all about the chain rule! Ok.
So we are going to do some simple problems involving the chain rule, just to get our
minds going on how the chain rule works, so that we can really understand
that implicit differentiation part. So here is the idea. When we have a function
inside of another function. Let's say I have my g function inside of f.
Then we want to take the derivative of the outside, so in this case I'm going to work on that
square root guy. Working on taking its derivative.
So let's see, one half, we'll reduce the power by one, negative one half.
So I've taken the derivative of the outside. Now we are going to leave that inside function
just as it is, and then multiply by the derivative of the inside.
Ok, so again in taking the derivative of our composition of functions, we have the derivative
of the outside, inside stays the same, multiplied by the derivative of the inside.
Now, it doesn't really matter what that inside function is.
In fact it could be a completely different function, like sin(x), and that same process is going
to play out. So let's do this problem again, now our inside
function is going to be sin(x). Alright, so just like last time we will do
the derivative of our outside function. So I took down the power, reduced it by one.
Inside function will stay the same. Then we'll multiply by the derivative of the inside.
So in this case the derivative of sin(x) is cos(x).
Alright, now again, I'm going to emphasize that it does not matter what that inside function is,
in fact it could be any general function, let's suppose it was just a g(x).
The process is not going to be any different. We'd start off with the derivative of the outside.
So there I've reduced the power.
Inside is going to stay exactly the same. And now I will multiply by the derivative
of the inside. Now, if you followed all three of those examples,
you can do implicit differentiation. What often messes people up
with implicit differentiation is that its a common practice to write that
inside function y(x) as simply just a y.
But really there is no difference in writing
it as y(x) or writing it just as y. So let's do a couple more chain rule problems,
and kind of practice with this notation. So here I have my f function and y is on the inside, y(x).
So if I'm taking the derivative of this guy,
I'd start with the derivative of the outside. cosine is the derivative of sine.
Our inside function is just going to stay exactly the same, y(x).
Then we'll multiply by the derivative of the inside.
So this is what the chain rule looks like when you are using that function notation.
Now since its common practice to simply write y, and assume that it is a function a x, here
is how that same problem looks, using the more common notation.
So I'm trying to take the derivative of f, and I start off taking the derivative of sine.
So there is my cosine. The inside is going to stay exactly the same,
so we're just going to write a y, multiplied by the derivative of the inside. y'
So that's implicit differentiation. I'm taking the derivative of my function, since I don't
know y exactly, it has to stay the same, and its derivative, I'm just going to express
that as y'. So now, we're actually ready to do some implicit differentiation problems.
And watch how that common notation, this y,
and the y' keep showing up every time I take the derivative.
So in this one I want to find the derivative of y with respect to x, and we are going to
assume that y is a function of x. So I'm going to start off by actually just
taking the derivative of both sides. And I'm using this notation to say, OK, I'm
going to take the derivative. And then we'll go ahead and do that in the next step.
Now, over here on the left, its that chain rule guy that's going to play a part.
So we'll do the derivative of the outside. You know, I've brought down the power, reduced
it by one. So this is to the power of one, so I'm not
going to write it. Inside will stay exactly the same, and then
I'll multiply by the derivative of the inside. There is all of that chain rule stuff playing
out. Now on the right side, the derivative of x,
with respect to x, is simply just one. So you'll do those derivatives like you normally would.
There's just a few more steps, usually we
want to isolate that derivative of y. So I'm gong to divide both sides by 2y.
And now I'm done. So y' = 1/(2y)
Let's give this another shot at something a little more complicated.
So again we want to find the derivative of y, with respect to x.
And we can see that we have a couple of y's hidden in there.
And we also have some x's, so when we do the derivative on those, it will be exactly like
a normally derivative. Think that we are using the chain rule, really
to take care of these guys. Alright, let's start off.
So I'm going to start off with the derivative of sin(y).
The derivative is cosine. Inside will stay exactly the same, multiplied
by the derivative of the inside, y' plus, now we are taking the derivative of
3x^2. 6x. And I'm going to use the chain rule again
to help me with this guy. So the derivative of the outside, I'd bring
down the 3, reduce it by a power of one. Now I'm multiplying by the derivative of the inside.
y' OK, off to the other side of the equal sign.
Derivative of 4 is simply 0. So I've been able to take the derivative,
I now have these derivative of y pieces in here, and we want to collect them together
and isolate them. So cos(y)y' - 3y^2y' the 6x let's go ahead
and put that one the other side. So -6x We have two of them so we are going to factor
those out front. So y'[cos(y) - 3y^2]
Alright, its looking pretty good, we have our y primes together, no we'll just divide
by all of this, and it will be completely isolated.
So y' = -6x/(cos(y) - 3y^2) So what it all boils down to what you are
doing with implicit differentiation, you're still finding the derivative of y with respect
to x, but what you see in the derivative is often need to know lots of information about
that function. For example, you need to know information
about x's and y values. Where as when you are doing just a normal
derivative, a function with just x's in it, then all you need to know is there x values
to go ahead and talk about that derivative. Alright, so remember that the implicit differentiation
is all about thinking about the chain rule, and you'll do just fine.
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