Question

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

Answer 1

`dy/dx = (2a)/(y)`

Explanation:

When we differentiate `y` wrt `x` we get `dy/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 `dy/dx`.

Example:

`d/dx(y^2) = d/dy(y^2)dy/dx = 2ydy/dx`

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

Now, we have:

`y^2=4ax \ \ \`, the equation of a Parabola in standard form

Implicitly differentiating wrt `x`

`d/dx y^2 = d/dx 4ax`

`:. 2ydy/dx = 4a`

`:. dy/dx = (4a)/(2y)`

`:. dy/dx = (2a)/(y)`