I'm working through Andrej Karpathy's awesome introduction to neural networks and the backpropagation algorithm, and am trying to differentiate the sigmoid function:

$$ \sigma(x) = \frac{1}{1+e^{-x}} $$

My understanding is the quotient rule holds that given an equation $y=\frac{t}{b}$ the derivative of the function should be:

$$y' = \frac{t'b - tb'}{b^{2}}$$

After applying the quotient rule to the sigmoid function, I thought the unsimplified result would be:

$$ \frac{-e^{-x}}{(1 + e^{-x})^{2}} $$

Because $t'$ is 0, which I thought would yield $ 0 - tb' $ in the numerator, or $-e^{-x}$. However, Karpathy's unsimplified derivative looks like:

$$ \frac{e^{-x}}{(1 + e^{-x})^{2}} $$

Does anyone know why the negative sign in the numerator gets dropped? I'd be very grateful for any help others can offer with this question!