# Differentiate y^2 = 4ax w.r.t x (Where a is a constant)?

Aug 2, 2017

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

#### Explanation:

When we differentiate $y$ wrt $x$ we get $\frac{\mathrm{dy}}{\mathrm{dx}}$.

However, we only differentiate explicit functions of $y$ wrt $x$. But if we apply the chain rule we can differentiate an implicit function of $y$ wrt $y$ but we must also multiply the result by $\frac{\mathrm{dy}}{\mathrm{dx}}$.

Example:

$\frac{d}{\mathrm{dx}} \left({y}^{2}\right) = \frac{d}{\mathrm{dy}} \left({y}^{2}\right) \frac{\mathrm{dy}}{\mathrm{dx}} = 2 y \frac{\mathrm{dy}}{\mathrm{dx}}$

When this is done in situ it is known as implicit differentiation.

Now, we have:

${y}^{2} = 4 a x \setminus \setminus \setminus$, the equation of a Parabola in standard form

Implicitly differentiating wrt $x$

$\frac{d}{\mathrm{dx}} {y}^{2} = \frac{d}{\mathrm{dx}} 4 a x$

$\therefore 2 y \frac{\mathrm{dy}}{\mathrm{dx}} = 4 a$

$\therefore \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{4 a}{2 y}$

$\therefore \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{2 a}{y}$