Derivative of Composite Function/Second Derivative

Theorem
Let $D_x^k u$ denote the $k$th derivative of a function $u$ $x$.

Then:
 * $D_x^2 w = D_u^2 w \left({D_x^1 u}\right)^2 + D_u^1 w D_x^2 u$

Proof
For ease of understanding, let Leibniz's notation be used:


 * $\dfrac {\mathrm d^k u} {\mathrm d x^k} := D_x^k u$

Then we have: