Find the derivative of $x^2+xy-y^2=1$ using the definition. Apply the definition of the derivative: $\displaystyle f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$. The function $f(x)$ is the function we want to differentiate, which is $x^2+xy-y^2=1$. Substituting $f(x+h)$ and $f(x)$ on the limit, we get

Cancel like terms $\left(x+h\right)^2+\left(x+h\right)y-y^2=1$ and $-x^2+xy-y^2=1$

Zero divided by anything is equal to zero

The limit of a constant is just the constant