I've been asked to do a few more implicit differentiation
problems, and this is something that I think confuses a lot of
people, so let's do a couple more problems.
And I think there was a typo.
I forgot the user that actually sent this, but I'm guessing
this is what he intended.
And if this isn't what he intended, this is something
close, so it should give the idea.
So the implicit function they gave was x squared minus 24xy
plus 16y squared minus 400x minus 300y is equal to 0.
So, how do we do this?
Well, we take the derivative of both sides with respect to x.
I could write the derivative operator on both sides, but I
think you know that at this point, so that's the next step.
So let's do that, and I'll do it in blue.
OK, derivative of x square with respect to x is 2x.
And now, let's see, it's going to be minus 24.
I'm going to take the constant term out, so minus 24 times
the derivative of xy.
So what's the derivative of xy?
We could use the product rule here.
The derivative of the first term x, well, that's
just 1, right?
So 1 is the derivative of x times the second expression,
or you could say the second mini-function or whatever,
times y, and then plus the first expression x times
the derivative of the second expression.
Well, what's the derivative of y with respect to x?
And this sometimes confuses people because it's so easy.
Well, it's just dy/dx.
Instead of writing dy/dx, I'm just going to write y prime
here just to get you used to a different type of notation,
although I prefer dy/dx because you realize you're
using the chain rule.
But anyway, xy prime, that's dy/dx, plus-- so we'll
do the chain rule again.
So let's take the derivative of this expression with respect to
y first, so that's just 32y.
But we're taking the derivative with respect to x, so we have
to take the derivative of this with respect y, which we did,
and then multiply that times the derivative of y with
respect to x, which is just dy/dx, or y prime.
And then this with respect to x, well, that's just minus 400.
And then what's the derivative of this with respect to x?
Well, it's minus 300, and then times the derivative of y with
respect to x because we're doing the chain rule, right?
Minus 300 is just the derivative of this
with respect y.
And then-- I want to hit this point.
You know, some people when they learn this, they just say, oh,
whenever I take a derivative of y, I just take the derivative
and I multiply by y prime.
And that's right.
That's essentially what you're doing.
But I want you understand that you're doing the chain
rule when you do that.
So you're taking the derivative with respect to y and then
multiplying that times the derivative with respect to x,
which we'll write as y prime now, and all of that
is equal to 0.
Now let's simplify this.
Let's see, I get 2x minus 24y minus 24xy prime plus 32yy
prime minus 400 minus 300y prime is equal to 0.
And let's switch colors.
So let's separate all the y prime terms out.
So let's see, what are the y prime terms?
We have this term has a y prime in it, we have this term has a
y prime in it, and we have this term has a y prime in it.
So we can write these.
Let's do the positive ones first.
So 32y, so this is the same thing as y prime.
y prime times 32y minus 24x minus 300.
Those are all the y prime terms, and then we're left
with just the regular terms.
So then plus 2x minus 24y minus 400 is equal to 0.
I just rearranged it and took out the y prime out of
this, this and this term.
Let's take all of these to the other side of the equation.
Now , essentially what I want to do is get all the terms that
don't have the y prime onto the right-hand side and then divide
by this, and then we will solve the equation.
Let's see, so we get y prime times-- and this is-- you could
also write, you know the derivative with respect
to x or dy/dx.
In general, even though it takes more time, I like to
normally write dy/dx, because, you remember, I'm taking the
derivative with respect to x. y prime?
I don't know.
You might be taking the derivative with respect
to who knows what?
Let's see, 32y minus 24x minus 300 is equal to-- put all these
on the right-hand side so you get-- I'll switch the order.
I like to put the positive numbers first.
24y minus 2x plus 400, right?
And then divide both sides by this, and I'll do it in a
different color because we're almost at the answer, and we
get y prime-- the derivative with respect to x is equal to
this thing-- 24y minus 2x plus 400, all of that over this
thing: 32y minus 24x minus 300.
And let's see, we could at least-- it looks like-- well,
everything's divisible by 2, so we could divide the numerator
and the denominator by 2, and you get 12y minus x plus 200
over 16y minus 12x minus 150.
And if I didn't make a careless mistake, that's the answer.
The derivative of y with respect to x any time is this
expression that has y's and x in it, and that's cool because
this is a fairly convoluted implicit function, but we can
figure out the slope of it at any point because we
know the derivative.
Let's do the next one.
And this one is-- this one looks simpler.
x minus y squared is equal to 8 times y minus 6.
All right.
So let's just take the derivative with respect to x of
both sides, and I'll use the dy/dx notation this time.
So let's first use the chain rule.
Let's take the derivative of the inside.
So what's the derivative of the inside function?
Well, we could do it either way.
Let's do it the other way.
I always do inside and out, but let's take the derivative of
this entire expression with respect to x minus y.
Well, that is 2 times x minus y.
And then it's going to be that times the derivative of x
minus y with respect to x.
So what the derivative of x minus y with respect to x?
Well, the derivative of x with respect to x is just 1.
And what's the derivative of-- well, we could say minus
y with respect to x.
Well, it's just minus dy/dx.
You could say the derivative of y with respect to y is-- or the
derivative of minus y with respect to y is minus 1, and
then by the chain rule, multiply that times by
the derivative of y with respect to x.
Either way, I think you get the point, hopefully.
So that's the derivative of the left-hand side.
And now what's the derivative of the right-hand side?
Well, we could do the chain rule or whatever, but, you
know, this is the same thing as 8y minus 48, right?
So we could rewrite this is 8y minus 48.
And so, if you take the derivative of this with
respect to x, the constant doesn't contribute.
So what's the derivative of 8y with respect to x?
Well, it's the derivative of 8y with respect to y, which is
just 8, and by the chain rule, times the derivative of 8y with
respect to y times the derivative of y
with respect to x.
And now, we just solve for dy/dx.
Let's see, you get-- and I will do a different color now.
2x minus 2y times 1 minus dy/dx is equal to 8 dy/dx.
And then what can we do?
We can distribute this out.
So 2x times this whole thing.
So 2x minus 2x dy/dx, right?
That times that, and then that times that, and
then this times this.
So minus 2y and then minus minus, or plus, 2y dy/dx.
And then let's take this because I want to put all the
dy/dx terms on the same side, so let's subtract this from
both sides of the equation.
So then we get minus 8 dy/dx is equal to 0.
And then the next step, well, let's just separate out
all the dy/dx terms.
So that's a dy/dx term, that's one, that's one.
They're easier to spot than when you write y prime.
So if we take those together and take out the dy/dx, we get
dy, the derivative of y with respect to x, times-- let's
see, are any of them positive?
I'll put that one first.
2y minus 2x, right?
That's this one.
Minus 8 and then plus 2x minus 2y is equal to 0.
Let's put these onto the right-hand side
of the equation.
We're almost there.
dy/dx times 2y minus 2x minus 8 is equal to 2y minus 2x-- I
just took those and put them on the right-hand side-- and
now divide both sides by this, and we are done.
dy/dx is equal to 2y minus 2x, all of that over
2y minus 2x minus 8.
And once again we can divide the numerator and the
denominator by 2.
So we're left with y minus x over y minus x minus 8.
And we're done.
Hopefully, you found that helpful.