Multivariable Chain Rule

The multivariable chain rule computes the total derivative of a variable z that depends indirectly on another variable t through intermediate variables x and y. Use it when z depends on x and y, both x and y depend on t, but t does not appear directly in z's expression. The rule states that dz/dt equals the partial derivative of z with respect to x times dx/dt, plus the partial derivative of z with respect to y times dy/dt.

Partial derivatives measure how z responds to changes in one variable while holding the other constant. In the multivariable chain rule, each partial derivative is multiplied by the rate of change of its corresponding intermediate variable. This combination captures the total effect of indirect changes on z. Understanding interpretations of partial derivatives helps clarify why each term contributes separately to the total derivative.

Taking a limit as the increment in t approaches zero transforms finite difference ratios into instantaneous rates of change. When the total change in z is divided by a small increment in t, the resulting ratios become derivatives only in the limit. This process ensures the chain rule captures the precise rate of change at a specific point rather than an average rate over an interval.

A weather balloon's temperature depends on altitude and humidity, both changing over time as the balloon rises. The total temperature change rate equals how temperature varies with altitude times the balloon's vertical speed, plus how temperature varies with humidity times the humidity's rate of change. This demonstrates that the total derivative combines multiple partial effects through their respective rates of change.

All functions involved must be differentiable for the multivariable chain rule to apply. This ensures that partial derivatives and ordinary derivatives exist and the limiting process is valid. Differentiability guarantees smooth, continuous behavior of the functions, allowing the decomposition of total change into partial components and their corresponding rates of change.

Direct dependence occurs when a variable appears explicitly in an expression; indirect dependence occurs when it influences the outcome through intermediate variables. In the multivariable chain rule, t does not appear directly in z's expression but still affects z through changes in x and y. Recognizing this distinction is essential for correctly applying the chain rule to composite multivariable functions.

The formula is dz/dt = (∂z/∂x)(dx/dt) + (∂z/∂y)(dy/dt). This expresses the total derivative of z with respect to t as the sum of two products: each partial derivative of z multiplied by the ordinary derivative of its corresponding intermediate variable. This structure generalizes to functions of three or more variables by adding additional terms following the same pattern.