Use chain rule and implicit differentiation to find dy/dx for y^3 = 1 + 3*x^2, then show that they are equal

When using the Chain Rule, it is best to put our equal in a form where there are no operations performed on the subject. i.e. y = (1 + 3*x²)^1/3

Then to help make it clear what we need to do, state the rule... and think about where we may use it; where 2 different functions are acting upon x, in the form...   f( g(x) ).

For equations in this form using the chain rule means.... dy/dx = g'(x) f' (g(x)). This means we differentiate 2 functions and multiply them together.

g(x) = 1 + 3*x²               and                 f(x) = ( g(x) )^1/3

If we differentiate these 2 functions we get...

g'(x) = 6x   and    f '(x) = 1/3 * ( g(x) )^-2/3     therefore.....      dy/dx = 1/3 * 6x * (1 + 3x²)^-2/3 = 2x(1 + 3*x²)^-2/3

When using implicit differentiation it is best we have our x terms and y terms on different sides. i.e. y³ = 1+ 3*x²

When performing implicit differentiation, we rewrite each side as an integral equation such as  y³ = 3*y² dy

Where the solution of the integral is same as our original terms.

3y² dy = 6x dx

Then rearranging the equation we can put our equation in the form dy/dx

dy/dx = (6x) / (3y²)

from here we need to rewrite y in terms of x.....  y = (1 + 3*x²)^1/3 and substitute this back into our equation

dy/dx = (6x) / (3(1 + 3*x²)^2/3)

We can then rearrange our solution into the form 

dy/dx = 2x*(1 + 3*x²)^-2/3