Multivariable Chain Rule

dz/dt = f_x dx/dt + f_y dy/dt — one term per path.

The idea

Theorem (Multivariable chain rule).

Let $z = f(x, y)$ be differentiable, and let $x$ and $y$ be differentiable functions of a single variable $t$. Then $\frac{dz}{dt} = \frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt}.$

In one variable the chain rule is a single product: a change in $t$ changes $x$, and the change in $x$ changes $f$. Here a change in $t$ reaches $f$ by two routes at once, through $x$ and through $y$, and the two effects add. Each term keeps the familiar shape — how sensitive $f$ is to an input, times how fast that input is changing — and there is exactly one term per input.

The rule extends to more variables by the same pattern. Draw a tree with $f$ at the top, its immediate inputs beneath it, and their inputs beneath those. Each path from $f$ down to the differentiation variable contributes the product of the derivatives along it, and the derivative is the sum over all such paths. Three intermediate variables give three terms; a longer chain gives longer products.

The symbol a derivative receives, $d$ or $\partial$, records how many variables the function being differentiated has: $dx/dt$ when $x$ depends on $t$ alone, $\partial x / \partial t$ when it depends on more.

Ways to work on it

• Walkthrough. Differentiate a function of two variables along a parameterized path with the chain rule.
• Practice. Evaluate dz/dt at a given t for a random f(x, y) along a small path.
• Hardest. Two-parameter version: compute f / s for f(x(s, t), y(s, t)).

Not sure where to start? Take the ten-question placement test.