Multivariable Chain Rule for Single Variable Function

Question

Suppose we have a function $$C(P, S(P))$$ that depends on $P$ and $S(P)$, where $S$ is another function.

I understand that from chain rule, we have \begin{align*} \frac{dC}{dP} &= \frac{\partial C}{\partial P} + \frac{\partial C}{\partial S}\frac{\partial S}{\partial P}. \end{align*}

However, what is the right way to interpret the difference between $\frac{dC}{dP}$ and $\frac{\partial C}{\partial P}$?

Answer 1

The difference here is that in $\dfrac{dC}{dP}$, $S$ is treated as a function of $P$, while in $\dfrac{\partial C}{\partial P}$, $S$ is treated as a constant.

Let us take an example to illustrate that these two ways really do give the same answer. Consider: $$C(P,S(P))=P\cdot S(P).$$ The single-variable product- and chain rules then give: $$\frac{dC}{dP}=S(P)+P\cdot S'(P).$$ But if we take partial derivatives as in the multivariable chain rule, we treat $S$ as a constant when differentiating with respect to $P$ and $P$ as a constant when differentiating with respect to $S$. We may write the function as just $C=PS$ to aid our thinking. We then get: $$\frac{dC}{dP}=\dfrac{\partial C}{\partial P}+\frac{\partial C}{\partial S}\frac{\partial S}{\partial P}=S+S\frac{dS}{dP}=S+P\cdot S'(P),$$ the same answer we got with the regular chain rule and product rule.

Answer 2

The difference is depending on how you express the function $C(P,S(P))$. If you expressed this in terms of both variables $P$ and $S$, you would be writing this in the format:

$$ C(P,S(P)) = \text{"expressions in $S$ and $P$"}. $$

Here the partial derivative $\frac{\partial C}{\partial P}$ just represents the partial derivative of expressions in $P$, holding variable $S$ constant, while the total derivative also differentiates $S$ to find the derivative with respect to the dependence of $S$ on $P$.

On the other hand, if substituted whatever expressions we have for $S = S(P)$ and simply write:

$$ C(P) \;\; =\;\; \text{"expressions solely in terms of $P$"} $$

then we can take the total derivative $\frac{dC}{dP}$ and this would agree with the total derivative that we found up above. So to summarize, if we express $C$ as a function in terms of $P$ and $S$, then $\frac{\partial C}{\partial P}$ just represents the derivative of $C$ with respect to $P$ but only for those terms that are not dependent on $S$. The total derivative $\frac{dC}{dP}$ is the total derivative, and this can be found by using the chain rule, or substituting the expression for $S$ with terms written in the variable $P$.