Different answers when differentiating implicitly vs explicitly $xy=1$

Question

$xy=1$ can be written as $y=\frac{1}{x}$. The derivative for $\frac{1}{x}$ is $-\frac{1}{x^2}$

But if I were to implicitly differentiate the original function $xy=1$, I get: $$\frac{dy}{dx} = -\frac{y}{x}$$ Where did I go wrong?

Answer 1

You have the right answer, just substitute your $y(x)$:

$$\frac{dy}{dx} = -\frac{y}{x}=-\frac{(\frac{1}{x})}{x}=-\frac{1}{x}\times\frac{1}{x}=-\frac{1}{x^2}= -y^2$$

Answer 2

By the product rule we get:

$$(xy)'=1'\Rightarrow x'y+y'x=0 \Rightarrow y+y'x=0 \Rightarrow y'=\frac{-y}{x} \tag{1}$$

But $\displaystyle xy=1\Rightarrow y=\frac{1}{x}$, replacing in (1), we get: $$y'=\frac{-1}{x^2}$$