1. ## Question about a rule for implicit differentiation

So, I'm given the equation:
(x^2+y^2)^2 = 4x^2y
The right side of the equation is easy enough, it requires the chain rule, so it becomes:
2(x^2+y^2)(2x+2y) right?
Now, the left side of the equation. My textbook lists it as 4x^2(dy/dx)+8xy.

My question is, is there some kind of rule for what's happening there, because I don't really understand? At first I tried to apply the product rule, but I guess that was wrong. If someone could clear up what exactly is happening there, that would be great!

2. The problem: (it took me few minutes to understand your problem)
$\displaystyle 4x^2y$
the derivative will be solved by using the product rule:
$\displaystyle 4*2x*y (derivative\ of\ x)+4x^2\frac{dy}{dx}$ $\displaystyle (derivative\ of\ y,\ so\ you\ have\ to\ write\ down\ \frac{dy}{dx})$
Thus, the result will be: $\displaystyle 8xy+4x^2\frac{dy}{dx}$

3. So, it IS the product rule. Well, I have no idea what I was doing haha. Thanks so much!

4. Originally Posted by Ohoneo
So, I'm given the equation:
(x^2+y^2)^2 = 4x^2y
The right side of the equation is easy enough, it requires the chain rule, so it becomes:
2(x^2+y^2)(2x+2y) right?
Now, the left side of the equation. My textbook lists it as 4x^2(dy/dx)+8xy.

My question is, is there some kind of rule for what's happening there, because I don't really understand? At first I tried to apply the product rule, but I guess that was wrong. If someone could clear up what exactly is happening there, that would be great!
1. Learn the difference between your right and left.

2. Using the product rule: $\displaystyle \frac{d}{dx}(4x^2y)=4 \left( \frac{d(x^2)}{dx}y+x^2\frac{dy}{dx} \right)$

CB