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## Calculus, all content (2017 edition)

### Unit 2: Lesson 30

Implicit differentiation introduction# Worked example: Implicit differentiation

AP.CALC:

FUN‑3 (EU)

, FUN‑3.D (LO)

, FUN‑3.D.1 (EK)

Implicit differentiation of (x-y)²=x+y-1. Created by Sal Khan.

## Want to join the conversation?

- At2:24, did Sal accidentally forget the - when distributing (2x - 2y) into (1 - (dy/dx))?(29 votes)
- Did Sal made a mistake? Should the answer should be the following?:

dy/dx = (2x - 2y - 1) / (2x - 2y + 1)

I've tried it twice and got the the same answer. Wolfram Alpha seems to confirm: http://www.wolframalpha.com/input/?i=derivative+with+respect+to+x+%28x-y%29%5E2+%3D+x+%2B+y+-+1.

Workings:

2(x - y)(1 - dy/dx) = 1 + dy/dx

=> (2x - 2y)(1 - dy/dx) = 1 + dy/dx

=> (2x - 2y) - (2x - 2y)dy/dx = 1 + dy/dx

=> dy/dx + (2x - 2y)dy/dx = 2x - 2y - 1

=> dy/dx(1 + (2x - 2y)) = 2x - 2y - 1

=> dy/dx = (2x - 2y - 1)/(2x - 2y + 1)(37 votes)- This is Sal's answer...

dy/dx = (2y - 2x + 1) / (2y - 2x -1)

If you change your answer like so...

dy/dx = (2x - 2y - 1) / (2x - 2y + 1)

dy/dx = -1(-2x + 2y +1) / -1(-2x + 2y -1)

dy/dx = (-2x + 2y +1) / (-2x + 2y -1)

dy/dx = (2y - 2x + 1) / (2y - 2x -1)

You can see that you and Sal both have the same answer.(2 votes)

- 2:11, how come we can simplify the left side of the equation by simply distributing (2x - 2y)? Do we need to FOIL it? Or would it give the same results?(22 votes)
- Way, way, way back in polynomials, Sal teaches one common alternative to the FOIL method that is called distribution. In this case, we have (2x - 2y)(1 - dy/dx). The method is to split one of the binomials into its two terms and then multiply each term methodically by the two terms of the second binomial. So, as he says, multiply (2x - 2y) times 1 and (2x - 2y) times -1(dy/dx) to get (2x - 2y) + (2y - 2x)dy/dx = 1 + dy/dx

As you noticed, the result is the same, and it should be. It is just another way to methodically multiply binomials.(14 votes)

- Hi everyone,

Quick Question - I'm stuck at3:40when Sal subtracts a dy/dx from the left hand side and gets -1dy/dx - how is this so?

In my mind dy/dx-dy/dx would somehow cancel eachother out but I think I've confused myself.(14 votes)- -dy/dx = -1 * dy/dx(3 votes)

- Shouldn't it be (2x-2y) - (2x-2y)(dy/dx) instead of (2x-2y)+(2x-2y)(dy/dx) at2:15(6 votes)
- He has switched the order of the variables, so that it is (2x-2y)+(2
**y**-2**x**)(dy/dx).

-(2x-2y) = 2y-2x(10 votes)

- At3:40i don't understand the process of subtracting the dy/dx(5 votes)
- The notation: dy/dx literally stands for the derivative of the function. Since it is an unknown you can treat it just as if it were a variable, which in a way, it is.

I hope this helps.(5 votes)

- Why is the d/dx multiplied by the constant ( in this case -1) equal to 0? Does it always equal 0 in every problem if it is a constant?(5 votes)
- There's a clue in the word 'constant'. d/dx multiplied by something is an operation to find this something's rate of change with respect to x. A constant, however, doesn't change at all, it stays constant! -1 constantly has the value -1. There are no changeable variables like x associated with it (like in, for example, -1x). So it has no rate of change. The rate of change of a constant always equals 0.(2 votes)

- At3:55, when Sal subtracts dy/dx from both sides, how does he end up with (2y-2x-1)dy/dx?(3 votes)
- (2y-2x) dy/dx - dy/dx = (2y-2x-1) dy/dx

It might help to imagine dy/dx as a single variable.

(2x - 2y)z - z = (2x - 2y - 1)z

If that doesn't help you may just want to expand and then re factor.

(2y-2x) dy/dx - dy/dx

2y dy/dx - 2x dy/dx - dy/dx Now factor out dy/dx

(2y - 2x - 1) dy/dx

Let me know if that didn't help.(7 votes)

- At2:08Sal decides to distribute (2x - 2y) and at2:26he distributes as -> (2x - 2y) + (2x - 2y)(1 - dx/dy).

My concern revolves around the distribution, that it should be -> (2x - 2y) - (2x - 2y)(1 - dx/dy) because (2x - 2y) distributes itself into (1 - dx/dy).

Why has Sal put a + ?(1 vote)- Sal's work here was correct but used a little sleight of hand that's easy to overlook. For the second part of the expression, instead of writing what you were expecting

- (2x - 2y)(1 - dy/dx)

he wrote

+ (2y - 2x)(1 - dy/dx)

Notice that he reversed the order of the terms in the first parentheses: instead of 2x - 2y he wrote 2y - 2x. That reversal of order is equivalent to multiplying the expression by -1, and that allows him to change the minus sign to a +.(9 votes)

- What does it mean to take a derivative of an equation? I know it tells the slope of the tangent line at any point of a function, ( if it's differentiable at that point). But an equation isn't a function... Right?(3 votes)
- A derivative is essentially a function of an original functon which gives the slope of the original function at every point. For example, the derivative of f(x)=2x would be the horizontal line y=2, since the slope of the original function at all points is 2.You can also find the derivative of other functions, and not just lines, but you wouldn't get a horizontal line derivative, since the slope would be constantly changing.(4 votes)

## Video transcript

Let's get some more practice
doing implicit differentiation. So let's find the derivative
of y with respect to x. We're going to assume
that y is a function of x. So let's apply our
derivative operator to both sides of this equation. So let's apply our
derivative operator. And so first, on
the left hand side, we essentially are just going
to apply the chain rule. First we have the
derivative with respect to x of x minus y squared. So the chain rule
tells us this is going to be the derivative
of the something squared with respect to the
something, which is just going to be 2 times x
minus y to the first power. I won't write the
1 right over there. Times the derivative of the
something with respect to x. Well, the derivative of x
with respect to x is just 1, and the derivative
of y with respect to x, that's what
we're trying to solve. So it's going to
be 1 minus dy dx. Let me make it a
little bit clearer what I just did right over here. This right over here
is the derivative of x minus y squared with
respect to x minus y. And then this right over
here is the derivative of x minus y with respect to x. Just the chain rule. Now let's go to the right
hand side of this equation. This is going to be equal to
the derivative of x with respect to x is 1. The derivative of y
with respect to x. We're just going to write
that as the derivative of y with respect to x. And then finally, the
derivative with respect to x of a constant, that's
just going to be equal to 0. Now let's see if we can
solve for the derivative of y with respect to x. So the most obvious thing to do. Let's make it clear. This right over here, I
can rewrite as 2x minus 2y. So let me do that so
I can save some space. This is 2x minus 2y If
I just distribute the 2. And now I can distribute
the 2x minus 2y onto each of these terms. So 2x minus 2y times 1 is
just going to be 2x minus 2y. And then 2x minus 2y
times negative dy dx, that's just going to be
negative 2x minus 2y. Or we could write that as
2y minus 2x times dy dx. Is equal to 1 plus dy dx. I'll do all my dy
dx's in orange now. 1 plus dy dx. So now there's a
couple of things that we could attempt to do. We could subtract 2x
minus 2y from both sides. So let's do that. So let's subtract 2x
minus 2y from both sides. So over here, we're going
to subtract 2x minus 2y from that side. And then we could also subtract
a dy dx from both sides, so that all of our dy dx's
are on the left hand side, and all of our non dy dx's
are on the right hand side. So let's do that. So we're going to subtract a
dy dx on the right and a dy dx here on the left. And so what are we left with? Well, on the left hand
side, these cancel out. And we're left with 2y minus
2x dy dx minus 1 dy dx, or just minus a dy dx. Let me make it clear. We could write this
as a minus 1 dy dx. So this is we can
essentially just add these two coefficients. So this simplifies to 2y minus
2x minus 1 times the derivative of y with respect
to x, which is going to be equal to-- on this
side, this cancels out. We are left with 1
minus 2x plus 2y. So let me write it that way. Or we could write
this as-- so negative, negative 2y is
just a positive 2y. And then we have minus 2x. And then we add that 1, plus 1. And now to solve
for dy dx, we just have to divide both sides
by 2y minus 2x minus 1. And we are left
with-- we deserve a little bit of a drum
roll at this point. As you can see, the
hardest part was really the algebra to solve for dy dx. We get the derivative
of y with respect to x is equal to
2y minus 2x plus 1 over 2y minus 2x minus 1.