An Introduction to Multivarible Chain Rule for Beginners

Multivarible chain rule is a good way to analyze the derivative of a machine learning model. In this tutorial, we will introduce it for machine learning beginners.

Multivarible Chain Rule

Let $z=f(x,y)$ , $x=g(t)$ and $y=h(t)$ , where $f$ , $g$ and $h$ are differentiable functions. Then $z=f(x,y)=f(g(t),h(t))$ is a function of $t$ , in order to compute the derivative $f$ with respect to $t$, we can use this formula:

You can understand it as follows:

Here is an example:

Let $z=x^y+x$ , where $x=sin(t)$ and $y=e^{5t}$ . Find $\frac{dz}{dt}$ using the chain rule.

Multivarible Chain Rule in diagram

We can understand multivarible chain rule using a diagram.

Vector in Multivarible Chain Rule

Rather than thinking of $x(t)$ and $y(t)$ as being separate functions, it’s common to package them together into a single, vector-valued function:

The multivarible chain rule will be: