Question

How do you find `dy/dx` by implicit differentiation of `x^2-y^3=0` and evaluate at point (1,1)?

Answer 1

`[dy/dx ]_{ (0,0) } =2/3`

Explanation:

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

However, we cannot differentiate a non implicit function of `y` wrt `x`. But if we apply the chain rule we can differentiate a function of `y` wrt `y` but we must also multiply the result by `dy/dx`.

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

We have:

`x^2-y^3=0`

Differentiate wrt `x`:

`2x-3y^2dy/dx=0 \ \ \ \` .... [1]

Although (in this case) we could find an implicit expression for `dy/dx` it is not always necessary, as we are asked to find the value at a particular point.

At `(1,1)` (noting that the point does indeed lie on the initial curve) we have (substituting into [1]):

`2-3dy/dx=0 => dy/dx=2/3`