Why am I getting three answers here for the same problem here?

Question

Its probably a silly question. But , why am i getting three different answers for the same question, just by differentiating by three different methods?

P.S: Please do forgive my handwriting, i don't know how to use latex yet.

Answer 1

Since $x^2y^2=1$, we must have $-xy^3=-\frac{1}{yx^3}=-\frac{y}{x}$. So, all the answers you got, are basically the same.

Answer 2

Since the implicit functions for this curve equation are simple enough to extract, you might also view the matter this way. The curve $ \ x^2y^2 = 1 \ $ can be expressed as $ \ y^2 \ = \ \frac{1}{x^2} \ \Rightarrow \ y \ = \ \pm \frac{1}{x} \ \ , $ which is the union of two "rectangular hyperbolas". The derivatives for these two explicit functions are then $ \ \frac{dy}{dx} \ = \ \mp \frac{1}{x^2} \ \ . $ This expression tells us that $ \ \frac{dy}{dx} < 0 \ $ over the entire domain of the hyperbola $ \ y = \frac{1}{x} \ $ and $ \ \frac{dy}{dx} > 0 \ $ over $ \ y = -\frac{1}{x} \ \ . $

With these derivatives in hand, and that $ \ xy = 1 \ $ or $ \ xy = -1 \ , $ respectively for the two hyperbolas, we can manipulate the derivative expressions in various ways. On $ \ xy = 1 \ \ , $ we then have everywhere

$$ \ \frac{dy}{dx} \ = \ - \frac{1}{x^2 · 1} \ = \ - \frac{1}{x^2 · xy} \ = \ - \frac{1}{yx^3} $$ $$ \text{or} \ \ \ \frac{dy}{dx} \ = \ - \frac{xy}{x^2 } \ = \ - \frac{y}{x} \ = \ - \frac{y·1^2}{x} \ = \ - \frac{y·(x^2y^2)}{x} \ = \ -xy^3 \ \ . $$

With the other hyperbola, the sign of the derivative and the sign of the product $ \ xy \ $ are reversed, so the same set of derivative expressions is produced (for instance, $ \ \frac{dy}{dx} \ = \ + \frac{1}{x^2 · 1} \ = \ + \frac{1}{x^2 · \ (-xy)} \ = \ - \frac{1}{yx^3} \ ) \ \ . $ We see that all of these versions are equivalent (and we could create any number of others as well) and that we may choose any of them to work with for the complete curve.