Total second derivative of function dependent on other functions

Question

We have the functions $A, B, C$ where $B=B(A)$, $C=C(B,A)$ and we want to compute the total second derivative of $C$, $C_{dd}$.

The first total derivative $C_d$ is given by the following formula:

$$ C_d = \sum_{i=1}^n \frac{\partial C}{\partial X_i} \cdot {X_i}_d = \frac{\partial C}{\partial A} \cdot A_d + \frac{\partial C}{\partial B} \cdot B_d $$ where the subscript $d$ indicates the total derivative.

Is there any general formula that gives the second order derivative $C_{dd}$ of a function with $n$ inputs?

Answer 1

Let $A=a(x), B=b(x),$ and $C=c(A,B)$

Then we have $\hspace{13ex}\dfrac{\mathrm d C}{\mathrm d x}=\dfrac{\partial C}{\partial A}\cdot\dfrac{\mathrm d A}{\mathrm d x}+\dfrac{\partial C}{\partial B}\cdot\dfrac{\mathrm d B}{\mathrm d x}$

So the second derivative is: $\dfrac{\mathrm d^2 C}{\mathrm d x~^2}=\dfrac{\mathrm d~~}{\mathrm dx}\left[\dfrac{\partial C}{\partial A}\cdotp\dfrac{\mathrm d A}{\mathrm d x}\right]+\dfrac{\mathrm d ~~}{\mathrm dx}\left[\dfrac{\partial C}{\partial B}\cdotp\dfrac{\mathrm d B}{\mathrm d x}\right]$

Now apply the Product Rule, then the Chain Rule, and clean up with a little algebra.

Then generalise.