# The Chain Rule

$[f(g(x))]_{′}=f_{′}(g(x))⋅g_{′}(x)$

In other words, $dxdy =dudy dxdu $

This is the basis for Backpropagation.

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.”*

### Multivariate Chain Rule

##### Chain Rule for Paths

The most basic form of the Chain Rule for multivariate calculus is written as $dtdz =dxdz dtdx +dydz dtdy $

When we have multiple variables, we simply draw chains with the relevant variables.