Total derivative w.r.t composite variable

Question

How to expanded the derivatives $$ \dfrac{\text{d}}{\text{d}C} f(A,B) $$ where $C=A+B$ ?

I'm looking for an expression involving e.g. $\partial/\partial A$ and $\partial/\partial B$.

Answer 1

When you write such an expression, you're implicitly asserting that $A(C)$ and $B(C)$ are functions of $C$, that happen to satisfy the relation $C = A(C) + B(C)$. Now you can apply the chain rule to get $$\frac{d}{dC}f(A,B) = d_1f(A,B)\frac{dA}{dC} + d_2f(A,B)\frac{dB}{dC}$$ where $d_if$ is the partial derivative of $f$ with respect to its $i$th argument.

Moreover you know that $$1 = \frac{dA}{dC} + \frac{dB}{dC}$$ and can use this relation to eliminate either $\frac{dA}{dC}$ or $\frac{dB}{dC}$, but generally speaking not both, unless the partial derivatives of $f$ are equal, $d_1f(A,B) = d_2f(A,B)$.