# How do you differentiate x^2+y^2=2xy?

Jul 16, 2015

Saikiran Reddy and Kwasi F. give excellent solutions to this. The answer, $\frac{\mathrm{dy}}{\mathrm{dx}} = 1$ might make us think about the question a bit.

#### Explanation:

For ${x}^{2} + {y}^{2} = 2 x y$, we get (by differentiating implicitly), $\frac{\mathrm{dy}}{\mathrm{dx}} = 1$.

That's the same as the derivative of a linear function with slope, $1$. Hmmmmm. Let's see:

If we have
${x}^{2} + {y}^{2} = 2 x y$

The we must also have:
${x}^{2} - 2 x y + {y}^{2} = 0$

Factoring gets us:
${\left(x - y\right)}^{2} = 0$

And the only way for that to happen is to have:
$x - y = 0$

So $y = x$

and $\frac{\mathrm{dy}}{\mathrm{dx}} = 1$.
(Which we already knew by differentiating, but this may be of interest as well.)