Multivariable Chain Rule

When a variable z depends on two intermediate variables, x and y, and both x and y vary with respect to a third variable t, the dependence of z on t is indirect. Although t does not explicitly appear in the expression for z, any change in t produces corresponding changes in x and y, which in turn alter the value of z. The objective is to determine the total derivative of z with respect to t, denoted as dz/dt.

Assuming that all functions involved are differentiable, the total change in z can be expressed as the sum of its partial changes with respect to x and y. Each partial change measures how z responds to variation in one variable while the other is held constant. When these changes are divided by a small increment in t and the limit is taken as the increment approaches zero, the resulting ratios become derivatives.

This process demonstrates that the total derivative of z with respect to t equals the derivative of x with respect to t multiplied by the partial derivative of z with respect to x, plus the derivative of y with respect to t multiplied by the partial derivative of z with respect to y. This relationship is known as the multivariable chain rule. It provides a systematic method for computing rates of change when variables are interconnected through intermediate dependencies.

A practical illustration is a weather balloon whose temperature depends on altitude and humidity. As the balloon ascends, both altitude and humidity vary over time. The total rate of change of temperature is therefore the sum of two contributions: the rate at which temperature changes with altitude multiplied by the balloon’s vertical speed, and the rate at which temperature changes with humidity multiplied by the rate at which humidity changes over time.