Chain Rule for Several Variables

Calculus 3 · Axiom Academy

LESSON Chain Rule for Several Variables Dependency flows down a tree — and the derivative is the sum of the contributions along every path to the variable you differentiate by. Put the function f at the top, the intermediate variables x and y in the middle, and the independent variable t at the bottom. Each branch carries a derivative; a path from f down to t records one route by which a change in t reaches f . Watch the tree grow, then the two routes light up. count the paths, multiply along each, add them up Suppose f(x,y) with x=g(t) and y=h(t) , so f is ultimately a function of t alone. The tree has two paths to t : and . Multiply along each, then add. Take f=x^ 2 +y^ 2 with and . The animation evaluates both paths at and adds them. 3. Several Independent Variables Now let x=g(s,t) and y=h(s,t) , so f depends on both s and t . Nothing new is needed: to get , sum only the paths that reach s ; for , sum the paths reaching t . Take f=xy with x=s^ 2 +t and y=s-t^ 2 . The animation selects the target s at the point (s,t)=(2,1) and sums its two paths. 4. Why It Works: Follow a Nudge The tree is not a mnemonic — it is the total differential . Nudge t by a small dt . That changes x by and y by ; those, in turn, change f by through the x branch and through the y branch. Watch the nudge ripple up both paths and add at f . Draw the tree: function on top, intermediate variables in the middle, independent variables at the bottom.

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