Definition of reciprocal derivative Suppose I define $y(x)=x^3$. Then, 
$$\frac{\mathrm dy(x)}{\mathrm dx}=3x^2.$$
however I don't see how $\displaystyle \frac{\mathrm dx(y)}{\mathrm dy}$=$\frac{1}{3x^2}$ because I never explicitly define $x(y)$ which has implications for coupling across equations of variables. $x(y)$ and $y(x)$ leads to a possible recursive definition $x(y(x))$ ... $y(x(y))$  Could someone help me resolve this?  Thank you!           
 A: What you see is abuse of notation, in a way. Here, $x$ and $y$ are both variables, but we have an equation relating them to each other, so we think of them as functions of each other.
If we define the function $f:\mathbb{R}\to\mathbb{R}$ such that $f(x)=x^3$, then it is possible to show that it has an inverse, $f^{-1}$. (At this point we don't care what the inverse actually is - we only need to know that it exists.)
Now if we have an equation of variables $y$ and $x$
$$
y=f(x),
$$
we can think of $y$ being a function of $x$, because it is uniquely determined by $x$ when this equation is satisfied. But on the other hand, this equation is equivalent to 
$$
x = f^{-1}(y),
$$
so we can think of $x$ being a function of $y$, even if we don't know the formula for $f^{-1}$. This does not lead to recursive deifinitions: now we know what the "function" $y$ is and what the "function" $x$ is.
The formula for the derivative of the inverse of $f$ is
$$
\frac{d f^{-1}(y)}{dy} = \left( f'(f^{-1}(y)) \right)^{-1}.
$$
Thinking of $x$ as a function of $y$, we get
$$
\frac{dx}{dy} = \frac{1}{3x^2}.
$$
A: Suppose $x$ is a differentiable function of $y$ such that $y=x^3$. We can differentiate both sides of this equation with respect to $y$ using the chain rule for the right-hand side to obtain $$1=3x^2\frac {dx}{dy}$$which gives the equation you state. This is a process known as implicit differentiation - $x$ is not given as an explicit function of $y$.
You are right to be cautious, because it is not always true that differentiable functions have an inverse. However assuming the inverse exists may lead to an explicit form, or to a contradiction. And this procedure may reveal key properties of the inverse.
For your recursive definitions note that $(x^{\frac 13})^3=x$
