# Geometric Definition of a Derivative

Before we get to the total derivative, what does the single-variable derivative mean? Well it’s a linear approximation at a given point. Now, imagine when we add a second variable, it’s no longer an approximation with a line, it’s a plane. Generalizing to higher dimensions, the derivative is a hyperplane approximation of a function.

# Mathematical Definition

**total derivative**. The equatino above doesn't help us figure out what it is at all. In order to start getting some intuition into what this mysterious function could be, let's look at an example.

# Total Derivative of a Linear Transformation

Let's look at an example. What's the derivative of $2x + 3y + 4z$? Well let's look at it like a matrix $$\begin{bmatrix}2 & 3 & 4\end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix}$$ The derivative is $\begin{bmatrix}2 & 3 & 4\end{bmatrix}$.

# Jacobian

# Chain Rule

David Witten