Find dy/dx by implicit differentiation: x^2y^2 - xy =4

Step-by-step explanation:

We will need to differentiate both sides.

Keep in mind the following:

* by power rule ()

* by power rule and chain rule.

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*

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  by product rule and some      already mentioned things. This is by product rule (.

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  by product rules and already mentioned things above.

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Let's put it all together by differentiating both sides:

I used the constant rule on the right hand side.

I also used difference rule. That is (f-g)'=f'-g'.

Now let's apply those mentioned things:

Distribute:

Put terms together:

Factor the out of the terms containing the :

Let's isolate the term containing the .

We will do this by adding on both sides and subtracting on both sides:

Now divide both sides the thing being multiplied by .

I really don't like the coefficient of not being in front so I'm going to rearrange that part using the commutative property of multiplication.

Let's see if this can be simplified.

I'm going to factor out what I can on both top and bottom.

The top terms contain a factor of .

The bottom terms contain a factor of .

We see that the other factor in top and bottom are opposite factors. I'm talking about the and the .

So if we factor out -1 on top we get:

Now we can cancel the common factor across the division there:

dy/dx = -y/x

Step-by-step explanation:

(x²)(2yy') + y²(2x) - [xy' + y] = 0

2x²yy' - xy' = y - 2xy²

y'(2x²y - x) = y - 2xy²

dy/dx = [y(1-2xy)]/[x(2xy-1)]

dy/dx = -y/x