How do you prove the chain rule?

Given the problem f(x)=(y+x)^5 we use the chain rule as a tool to find the derivative. However, it is difficult to use effectively in further problems without understanding the mechanics of the function. F(x) can be broken down into two functions where h(g(x)) is a composite function equalling f(x). H(u)=u^5 and g(x)=y+x.Typically we write derivatives in prime notation, for example f'(x), but it is more useful in this case to write it in operator notation. Operator notation allows us to break the derivative down further into smaller components the differentials dy and dx and treat it as a fraction as dy/dx.F'(x)=dy/dxH'(u)=dy/duG'(x)=du/dxIf we multiply dy/dudu/dx we can apply fraction multiplication rules of the numerator multiplied by the other numerator and the denominator by the other denominator and we can rearrange them in any order. So that:dy/dudu/dx=(dydu)/(dxdu)=du/dudy/dxAs du/du=1dy/dudu*dx=dy/dxH'(u)*g'(x)=dy/dx thus we have proven the chain rule.