Editing Chain rule (section)

==Intuitive explanation==
Intuitively, the chain rule states that knowing the instantaneous rate of change of {{math|''z''}} relative to {{math|''y''}} and that of {{math|''y''}} relative to {{math|''x''}} allows one to calculate the instantaneous rate of change of {{math|''z''}} relative to {{math|''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."<ref>[[George F. Simmons]], ''Calculus with Analytic Geometry'' (1985), p. 93.</ref>

The relationship between this example and the chain rule is as follows. Let {{mvar|z}}, {{mvar|y}} and {{mvar|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 <math DISPLAY = inline>\frac {dz}{dy}=2.</math> Similarly, <math DISPLAY = inline>\frac {dy}{dx}=4.</math> So, the rate of change of the relative positions of the car and the walking man is
<math display="block">\frac{dz}{dx}=\frac{dz}{dy}\cdot\frac{dy}{dx}=2\cdot 4=8.</math>

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,
<math display="block">\frac{dz}{dx}=\frac \frac{dz}{dt}\frac{dx}{dt},</math>
or, equivalently,
<math display="block">\frac{dz}{dt}=\frac{dz}{dx}\cdot \frac{dx}{dt},</math>
which is also an application of the chain rule.