The Multivariable Chain Rule
Differentiate a composition when several inputs each depend on other variables, by summing one contribution per path through the dependency diagram.
What a learner can do afterwards
- Draw the dependency diagram and read the terms off it
- Differentiate z = f(x, y) with x and y both functions of t
- Handle two intermediate variables and two independent ones
1 · Read
The multivariable chain rule says every route from the outer variable to t contributes one product term. Draw the dependency diagram first: z on top, x and y in the middle, t at the bottom. Each path gives an outer partial times an inner derivative, and you add the paths.
Take z = x squared + y squared with x = t and y = 2t. Substituting gives z = 5t squared, so dz/dt = 10t, which is 10 at t = 1. Or take z = 3x + 4y with x = t squared and y = t: dz/dt = 3(2t) + 4 = 6t + 4, which is 16 at t = 2.
With two independent variables you repeat the same diagram once per target. For z = f(x, y) with x(s, t) and y(s, t), hold t fixed: dz/ds = (dz/dx)(dx/ds) + (dz/dy)(dy/ds). Same picture, new target at the bottom.
The single variable chain rule is the one path case: derivative of the outside times derivative of the inside. Every multivariable term has exactly this shape. If a tree feels messy, shrink it to this case first.
Draw the diagram, write one product per path, and add them.
2 · Watch
Take it off screen
Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.