Why am I getting two slopes on differentiating the standard parabola equation?

Question

$y^2=4ax$

$y= \pm 2 \sqrt {ax} $

Now, we have two cases. And after differentiating both we get two different slopes which is not possible.

$\frac {dy}{dx} = \pm \frac {2a}{y}$

What am I doing wrong?

Answer 1

Be careful $$y= \pm 2 \sqrt {ax}\implies \frac {dy}{dx} = \pm \frac {2a}{2 \sqrt {ax}} =\pm \frac {2a}{\pm y} = \frac {2a}{ y} $$

Your mistake was in substitution of $ 2 \sqrt {ax}$ for $y$