Thread: Q about using the product rule with implicit differentiation

[edit] ignore the bit about the product rule in the title [edit]


Hi everyone, this is my first time posting here I hope you guys can help me.
I am trying to get the derivative of x^2 - xy + y^2 = 3

After the first step I got:
2x - y - x(dy/dx) + 2y(dy/dx) = 0

When I get to the step where I would move all like terms to one side, I could either move all non(dy/dx) terms to the right hand side and have the equation
-x(dy/dx) + 2y(dy/dx) = y - 2x

OR if I were to move the (dy/dx) terms to the right hand side I would get:
2x - y = x(dy/dx) - 2y(dy/dx)

The first route will give me the derivative (y - 2x)/(2y - x)
The second route will give me derivative (2x - y)/(x - 2y)

It seems like I attacked both possible routes correctly, but I'm pretty sure one of the final derivatives is incorrect because (y - 2x)/(2y - x) does not equal (2x - y)/(x - 2y).... or do they?

feedback would be much appreciated!!

Your work is correct.

Originally Posted by robo_robb

...but I'm pretty sure one of the final derivatives is incorrect because (y - 2x)/(2y - x) does not equal (2x - y)/(x - 2y).... or do they?

They are equal. Factor a -1 out of the numerator and denominator:









It is helpful to remember that, in general,

Ohh, I see now. Thanks a bunch!

Originally Posted by Reckoner

It is helpful to remember that, in general,

And it is also helpful to remember that, if this fact was necessary in the homework, it will almost certainly be necessary on the next test!