quality:corruption, structural:scholar_edition
Intuitive explanation

Intuitively, the chain rule states that knowing the instantaneous rate of change of z relative to y and that of y relative to x allows one to calculate the instantaneous rate of change of z relative to x as the product of the two rates of change.

As put by George F. Simmons: "If a car travels twice as fast as a bicycle and the bicycle is four times as fast as a walking man, then the car travels 2 × 4 = 8 times as fast as the man."

The relationship between this example and the chain rule is as follows. Let z, y and x be the (variable) positions of the car, the bicycle, and the walking man, respectively. The rate of change of relative positions of the car and the bicycle is d z d y = 2. {\textstyle {\frac {dz}{dy}}=2.} Similarly, d y d x = 4. {\textstyle {\frac {dy}{dx}}=4.} So, the rate of change of the relative positions of the car and the walking man is

The rate of change of positions is the ratio of the speeds, and the speed is the derivative of the position with respect to the time; that is,

or, equivalently,

which is also an application of the chain rule.