Multivariable Chain Rule

Consider a variable z that depends on x and y, while both x and y depend on another variable t.

Although t does not appear directly in the expression for z, variation in t alters x and y, resulting in an indirect change in z. Here, the goal is to find the total derivative, dz/dt.

Assuming all functions are differentiable, the total change in z is the sum of its partial changes with respect to x and y.

By dividing this relationship by Delta t and taking the limit as Delta t approaches zero, the ratios transform into derivatives.

This shows that the total derivative is the sum of the derivatives of x and y with respect to t, each multiplied by the corresponding partial derivative of the original function, giving the Multivariable Chain Rule.

For example, consider a weather balloon whose temperature depends on both altitude and humidity. As the balloon rises, both these variables change over time.

Using the chain rule for multivariable functions, the total temperature change is the sum of: how temperature varies with altitude times the balloon's speed, plus how the temperature varies with humidity times the humidity's rate of change over time.