This video is all about implicit differentiation, which is a method that we use to find the
This video is all about implicit differentiation, which is a method that we use to find the

derivative of implicit functions.
derivative of implicit functions.

Now you're probably wondering what it even means for a function to be called "implicit".
Now you're probably wondering what it even means for a function to be called "implicit".

Well, the word "implicit" is the opposite of "explicit", and the word "explicit" basically
Well, the word "implicit" is the opposite of "explicit", and the word "explicit" basically

means "clearly defined or expressed".
means "clearly defined or expressed".

So explicit functions are the ones you've already seen a lot of in your calculus class.
So explicit functions are the ones you've already seen a lot of in your calculus class.

They look like this: a dependent variable, in most cases we call this "y", expressed
They look like this: a dependent variable, in most cases we call this "y", expressed

as the function of an independent variable, usually "x".
as the function of an independent variable, usually "x".

These types of functions can be differentiated using our usual methods without any trouble.
These types of functions can be differentiated using our usual methods without any trouble.

But if a function is not explicit, that is, if y is not clearly expressed as the function
But if a function is not explicit, that is, if y is not clearly expressed as the function

of x, we call it an implicit function.
of x, we call it an implicit function.

Implicit functions look like this: x's and y's are tangled up together in such a way
Implicit functions look like this: x's and y's are tangled up together in such a way

that they are not easily separated.
that they are not easily separated.

In this function, for example, we can try to isolate the variables a couple of different
In this function, for example, we can try to isolate the variables a couple of different

ways, but it just doesn't work.
ways, but it just doesn't work.

This is where implicit differentiation comes into play.
This is where implicit differentiation comes into play.

Let's go ahead and use it to differentiate this function.
Let's go ahead and use it to differentiate this function.

The first thing we'll do is take the derivative of both sides with respect to x.
The first thing we'll do is take the derivative of both sides with respect to x.

Notice that the right side of the equation is a constant, and just like always, the derivative
Notice that the right side of the equation is a constant, and just like always, the derivative

of a constant will always be zero.
of a constant will always be zero.

On the left side, we'll need to take the derivative of both components, x^2 and y^2.
On the left side, we'll need to take the derivative of both components, x^2 and y^2.

The x^2 component is easy; we just use power rule to find that the derivative of x^2 is
The x^2 component is easy; we just use power rule to find that the derivative of x^2 is

2x.
2x.

And then we come to the y^2, and it seems pretty weird.
And then we come to the y^2, and it seems pretty weird.

How can we differentiate y^2 with respect to x?
How can we differentiate y^2 with respect to x?

Here's what implicit differentiation tells you to do: first, take the derivative as normal,
Here's what implicit differentiation tells you to do: first, take the derivative as normal,

so here we'll use the power rule again to get 2y.
so here we'll use the power rule again to get 2y.

Then, we need to multiply this term by dy/dx.
Then, we need to multiply this term by dy/dx.

The reason for this extra multiplication may seem a little mysterious right now, but we'll
The reason for this extra multiplication may seem a little mysterious right now, but we'll

talk about how this is actually just the chain rule in disguise in just a minute.
talk about how this is actually just the chain rule in disguise in just a minute.

Once you have the entire function differentiated with respect to x, we just need to use algebra
Once you have the entire function differentiated with respect to x, we just need to use algebra

to solve for dy/dx, which remember is the derivative of y with respect to x.
to solve for dy/dx, which remember is the derivative of y with respect to x.

This is the question we've been working to solve all along: in this case, we've found
This is the question we've been working to solve all along: in this case, we've found

that the derivative of y with respect to x is -x/y.
that the derivative of y with respect to x is -x/y.

Okay, so now that we've seen that this method works, let's take a closer look at why it
Okay, so now that we've seen that this method works, let's take a closer look at why it

works.
works.

Why do we multiply some parts of the function by dy/dx?
Why do we multiply some parts of the function by dy/dx?

To start to answer this question, let's look at the implicit function we started with.
To start to answer this question, let's look at the implicit function we started with.

It's really important to understand that one of these variables is the function of the
It's really important to understand that one of these variables is the function of the

other.
other.

This is the "implication" of the implicit function.
This is the "implication" of the implicit function.

Usually, we use x as the independent variable and y as the dependent variable, so y is actually
Usually, we use x as the independent variable and y as the dependent variable, so y is actually

a function of x.
a function of x.

We could even rewrite the function as x^2+[f(x)]^2 in order to represent this.
We could even rewrite the function as x^2+[f(x)]^2 in order to represent this.

Let's keep in mind this substitution.
Let's keep in mind this substitution.

The function now looks like this, x^2+[f(x)]^2.
The function now looks like this, x^2+[f(x)]^2.

But that function of x^2 now looks like a function inside of a function, or a composite
But that function of x^2 now looks like a function inside of a function, or a composite

function, which calls for the chain rule.
function, which calls for the chain rule.

So let's go ahead and apply the chain rule here.
So let's go ahead and apply the chain rule here.

We'll take the derivative of the outside function, 2f(x), and multiply it by the derivative of
We'll take the derivative of the outside function, 2f(x), and multiply it by the derivative of

the inside function, which we can denote as f'(x).
the inside function, which we can denote as f'(x).

And if we substitute y back in for f(x), and denote the derivative of f(x) as dy/dx, you
And if we substitute y back in for f(x), and denote the derivative of f(x) as dy/dx, you

can see that we end up with the derivative we got using implicit differentiation.
can see that we end up with the derivative we got using implicit differentiation.

So, implicit differentiation is just a glorified application of the chain rule!
So, implicit differentiation is just a glorified application of the chain rule!

Let's do a quick review of the steps we took to use implicit differentiation.
Let's do a quick review of the steps we took to use implicit differentiation.

First, we differentiate both sides of the equation with respect to the independent variable.
First, we differentiate both sides of the equation with respect to the independent variable.

Whenever we get to a term that contains the dependent variable, we recognize it as a function
Whenever we get to a term that contains the dependent variable, we recognize it as a function

of the independent variable and make sure to multiply it by the derivative with respect
of the independent variable and make sure to multiply it by the derivative with respect

to x.
to x.

And finally, we use algebra to solve for the derivative.
And finally, we use algebra to solve for the derivative.

It's as simple as that.
It's as simple as that.