Theory

Chain rule over a curve

Suppose $f(x,y)$ is a real-valued function on the plane, and $\mathbf{r}(t)=(x(t),y(t))$ is a parametric curve. There is a recipe to compute the derivative of the composite function $(f\circ \mathbf{r})(t)=f(x(t),y(t))$:

$$\frac{d}{dt}f(x,y)=\frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{dy}{dt}.$$

Chain rule over a curve (part 2)

Derivation of chain rule over a curve

Let’s suppose that some small change $\Delta t$ produces a change $\Delta x$ and $\Delta y$ via the parametrization $\mathbf{r}(t)$. By differentiability of $f$, we have:

$$\Delta f=\frac{\partial f}{\partial x}\Delta x+\frac{\partial f}{\partial y}\Delta y+\Delta x\cdot {\epsilon }_1+\Delta y\cdot {\epsilon }_2.$$

Divide both sides by $\Delta t$, and take the limit as $\Delta t\to 0$. Notice that ${\epsilon }_1\to 0$ and ${\epsilon }_2\to 0$ because $\Delta t\to 0$ implies $\Delta x,\Delta y\to 0$. It follows that we have:

$$\frac{\Delta f}{\Delta t}\to \frac{df}{dt}=\frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{dy}{dt}+\frac{dx}{dt}\cdot 0+\frac{dy}{dt}\cdot 0.$$

So the sum of products in this chain rule comes from the linear approximation formula, which in turn comes from the tangency of the trace curves.

Chain rule over several parameters

The chain rule over a curve can be used to derive a more general chain rule for the case when the input variables are functions of several parameters. The formulas can be remembered as “sum of all products” (considering all products that make sense).

Suppose $f(x,y)$ is a function, and suppose $x(s,t)$ and $y(s,t)$ are both functions of $s$ and $t$. We can “consider” $f$ itself to be a function of $s$ and $t$ via the composition. Then:

$$\begin{array}{rl}\frac{\partial f}{\partial s}& =\frac{\partial f}{\partial x}\frac{\partial x}{\partial s}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial s},\\ \frac{\partial f}{\partial t}& =\frac{\partial f}{\partial x}\frac{\partial x}{\partial t}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial t}.\end{array}$$

These formulas may be derived immediately from the formula for the chain rule over a curve by considering the trace curves: let $t=b$ remain fixed for the first equation, and let $s=a$ remain fixed for the second equation.

Chain rule over several parameters (part 2)

It is easy to generalize the chain rule to an arbitrary number of variables and parameters. Formally, the rule is:

$$\frac{\partial f}{\partial t_i}=\frac{\partial f}{\partial x_1}\frac{\partial x_1}{t_i}+\frac{\partial f}{\partial x_2}\frac{\partial x_2}{t_i}+\cdots +\frac{\partial f}{\partial x_n}\frac{\partial x_n}{t_i},$$

where $f(x_1,x_2,\dots ,x_n)$ is a function of many variables, and each $x_{\ell }(t_1,t_2,\dots ,t_k)$ is a function of many parameters.

To remember the formula, it sometimes helps to draw a tree. The summation has one term for each path from the output variable $z=f(x,y)$ down to the relevant input variable: