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Thank you for taking my question. I am homeschooling my son in Calculus, but for some reason, I can't get this problem correct (which makes it hard to teach it); my teaching book just isn't very helpful. I was hoping you could show me the correct answer and how you got it.
Thanks, John

The question is:
What is the slope of the tangent to the curve 3y^(2)-2x^(2)=6-2xy at the point (3,2)

The question asking for a slope, which is supposed to be dy/dx (and not dx/dy)

(in fact it is arbitrary to take x as independent variable and y the dependent, for example y = 2x + 1 can be rewritten as x = (y - 1)/2, so, it's a matter of perspective...ok i'm digressing a bit XD)

Your problem lies in your unease with an implicit function (i.e. u can't put y only to the left of = sign, while the x to the right) The way to deal with this is using chain rule, and differentiate term by term w.r.t. x

Let's look at the first term, 3y^2. It's expressed in terms of y, yet we need to differentiate w.r.t x, how?
Using chain rule: dz/dx = dz/dy * dy/dx, here z = 3y^2, and then u should get dz/dy = 6y, and therefore dz/dx = 6y*dy/dx   (dun touch dy/dx, we will deal with it later)

Just like we can add/substract/divide/multiply LHS and RHS at the same time, we need to differentiate every term to make sure the = sign holds, so the next target is -2x^2, which u should be able to get -4x

The RHS is simple, 6 disappears and d(-2xy)/dx need to use the product rule: d(y1*y2)/dx = y1*dy2/dx + y2*dy1/dx, so we have d(-2xy)/dx = y*d(-2x)/dx + (-2x)*dy/dx = -2y -2x*dy/dx
(Here I decomposed -2xy into -2x and y, ofc u can make it anyway u like, such as -y and 2x, but notice xy and -2 won't work, why?)

Now we get a new equation of dy/dx, which is 6y*dy/dx - 4x = -2y -2x*dy/dx, now put dy/dx all to left,
we have (6y+2x)*dy/dx = 4x - 2y  => dy/dx = (4x-2y)/(6y+2x)
sub in ur values, at (3,2) the gradient of the tangent is 4/9

(**What if we ask for the gradient at a point where 6y+2x is zero? dy/dx is undefined? no, it will be infinite, meaning it is vertical! Think about it ;P)

Verify if my result is correct. (too tired, need to go sleep)

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