Main content

## Implicit differentiation

# Showing explicit and implicit differentiation give same result

AP.CALC:

FUN‑3 (EU)

, FUN‑3.D (LO)

, FUN‑3.D.1 (EK)

## Video transcript

What I want to show
you in this video is that implicit
differentiation will give you the same result as,
I guess we can say, explicit
differentiation when you can differentiate explicitly. So let's say that I
have the relationship x times the square root
of y is equal to 1. This one is actually
pretty straightforward to define explicitly in
terms of x, to solve for y. So if we divide both sides
by x, we get square root of y is equal to 1/x. And then if you square both
sides, you get y is equal to 1 over x squared, which
is the same thing as x to the negative 2 power. And so if you want
the derivative of y with respect to x, this
is pretty straightforward. This is just an application
of the chain rule. So we get dy dx is
equal to negative 2 x to the negative 2 minus 1--
x to the negative 3 power. So that's pretty
straightforward. But what I want to see is if
we get the same exact result when we differentiate
implicitly. So let's apply our
derivative operator to both sides of this equation. And so let me make
it clear what we're doing-- x times the square root
of y and 1 right over there. When you apply the
derivative operator to the expression on
the left-hand side, well, actually, we're going
to apply both the product rule and the chain rule. The product rule
tells us-- so we have the product of
two functions of x. You could view it that way. So this, the product
rule tells us this is going to be the
derivative with respect to x of x times the
square root of y plus x, not taking its derivative,
times the derivative with respect to x of
the square root of y. Let me make it
clear, this bracket. And on the right-hand
side, right over here, the derivative with respect
to x of this constant, that's just going
to be equal to 0. So what does this simplify to? Well, the derivative with
respect to x of x is just 1. This simplifies to
1, so we're just going to be left with the square
root of y right over here. So this is going to simplify
to a square root of y. And what does this
over here simplify to? Well the derivative with respect
to x of the square root of y, here we want to
apply the chain rule. So let me make it clear. So we have plus this x plus
whatever business this is. And I'm going to
do this in blue. Well, it's going to be the
derivative of the square root of something with respect
to that something. Well, the derivative
of the square root of something with respect
to that something, or the derivative of
something to the 1/2 with respect to that something,
is going to be 1/2 times that something to the
negative 1/2 power. Once again, this right
over here is the derivative of the square root of
y with respect to y. We've seen this multiple times. If I were to say the derivative
of the square root of x with respect to x, you would
get 1/2 x to the negative 1/2. Now I'm just doing it with y's. But we're not done yet. Remember, our
derivative operator wasn't to say with respect to y. It's with respect to x. So this only gets us
with respect to y. We need to apply the
entire chain rule. We have to multiply that
times the derivative of y with respect to x in order
to get the real derivative of this expression
with respect to x. So let's multiply
times the derivative of y with respect to x. We don't know what that is. That's actually what
we're trying to solve for. But to use the
chain rule, we just have say it's the derivative
of the square root of y with respect to y times
the derivative of y with respect to x. This is the derivative of
this thing with respect to x. So we get this on
the left-hand side. On the right-hand
side, we just have a 0. And now, once again,
we can attempt to solve for the derivative
of y with respect to x. And maybe the
easiest first step is to subtract the square
root of y from both sides of this equation. And actually, let me move
all of this stuff over, so I have, once again,
more room to work with. So let me cut it, actually. And then let me paste it. Let me move it over,
right over here. So we went from there to there. I didn't gain a
lot of real estate, but hopefully this
helps a little bit. And actually, I
don't even like that. Let me leave it
where it was before. So then, if we subtract
the square root of y from both sides-- and
I'll try to simplify as I go-- we get this thing,
which I can rewrite as x times-- well, it's just going
to be x in the numerator divided by 2 times the square root
of y. y to the negative 1/2 is just the square root
of y in the denominator. And 1/2, I just put the
2 in the denominator there-- times dy dx
times the derivative of y with respect to x
is going to be equal to the negative
square root of y. I just subtracted the square
root of y from both sides. And actually, this is
something that I might actually want to copy and paste up here. So copy and then paste. So let's go back up here, just
to continue our simplification solving for dy dx. Well, to solve
for dy dx, we just have to divide both sides by
x over 2 times the square root of y. So we're left with dy dx--
or dividing both sides by this is the same
thing as multiplying by the reciprocal of
this-- is equal to 2 times the square root of y
over x-- over my yellow x-- times the negative
square root of y. Well, what's this
going to simplify to? This is going to be
equal to y times-- the square root of y times the
square root of y is just y. The negative times the
2, you get negative 2. So you get negative 2y over x
is equal to the derivative of y with respect to x. Now you might be
saying, look, we just figured out the
derivative implicitly, and it looks very different than
what we did right over here. When we just used the power
rule, we got negative 2 x to the negative third power. The key here is to realize
that this thing right over here we could solve explicitly
in terms of-- we could solve for y. So we could just make
this substitution back here to see that these
are the exact same thing. So if we make the substitution
y is equal to 1 over x squared, you would get dy dx, the
derivative of y with respect to x, is equal to our negative
2 times 1 over x squared, and then all of that over x,
which is equal to negative 2 over x to the third, which is
exactly what we have over here, negative 2 x to the
negative third power.

AP® is a registered trademark of the College Board, which has not reviewed this resource.