Chain Rule in Several Variables

The central idea

If $f:\mathbb R^m\to\mathbb R^p$ is differentiable at $g(a)$ and $g:\mathbb R^n\to\mathbb R^m$ is differentiable at $a$, then $D(f\circ g)(a)=Df(g(a))Dg(a)$, a $p\times n$ matrix. For scalar $z=f(x(t),y(t))$, this becomes $dz/dt=f_xx'+f_yy'$.

Definitions, hypotheses, and notation

Differentiability, not merely existence of partial derivatives, is the correct hypothesis for the multivariable chain rule. Continuous partial derivatives near the point are a common sufficient condition. In coordinate changes, the Jacobian maps small input displacements to small output displacements; its determinant instead measures local signed volume scaling when square.

For a scalar outer function, $Df$ is a row matrix under one convention while the gradient is a column vector. The formula $d(f\circ g)/dt=\nabla f(g(t))\cdot g'(t)$ avoids ambiguity, but matrix products must retain a consistent convention.

Conceptual meaning

The derivative is a linear map, and composing functions composes their best local linear maps in reverse reading order. Matrix dimensions enforce which variables depend on which; a dependency tree is the scalar version of the same multiplication.

A dependable method and decision rule

1. List intermediate and independent variables with their dimensions.
2. Evaluate outer partial derivatives at the current intermediate point.
3. Compute derivatives of each intermediate variable.
4. Sum over every dependency path or multiply Jacobians in compatible order.
5. Check the final derivative shape matches output dimension by input dimension.

Fully worked example

Graphical or geometric meaning

A dependency diagram has arrows $t\to x$, $t\to y$, and both $x,y\to z$. Each route contributes a product of derivatives; summing the two routes gives the total rate of change.

Common mistakes and why they fail

Verification and reasonableness checks

• Write matrix dimensions beside each Jacobian.
• Substitute intermediate formulas and differentiate directly when feasible.
• Verify units along each dependency path agree before adding.

Draw the dependency graph first

When $z=f(x,y)$ and both $x$ and $y$ depend on $t$, every route from $t$ to $z$ contributes: $dz/dt=f_x\,dx/dt+f_y\,dy/dt$. With several intermediate variables, a dependency tree or Jacobian product prevents omitted paths. Evaluate each partial derivative at the current intermediate point, not at the raw input. In matrix form, derivative matrices multiply in composition order, and their dimensions provide an immediate check. Holding a variable constant for a partial derivative is different from declaring it independent along the composed path. After computing, test a simple direction or expand the composite formula directly when feasible; both methods should produce the same total derivative.

Practice

1. If $z=x^2+y^2$, $x=t$, $y=t^2$, find $dz/dt$.
2. What is the shape of $Df$ for $f:\mathbb R^3\to\mathbb R^2$?
3. For $z=f(r\cos\theta,r\sin\theta)$, find $z_r$.

Connections and next steps

• Partial Derivatives
• Gradient
• Directional Derivatives