Prerequisite concepts
The derivative is a ratio of small changes
Difference / Change
The derivative relates how much $f$ changes as $x$ changes
The derivative can be approximated by the slope of a secant line
The derivative is a limit
The derivative is a function
The derivative of a constant is zero
The derivative is a linear operator
Power law
The derivative at a cusp is undefined
Variables can be held constant
Product rule
Single variable chain rule
You can flip a derivative
"With respect to what" matters
The magnitude of the derivative at a point is the slope of a tangent line at that point
One can view $\frac{df}{dx}$ as approximately given by a fraction where $df$ is a small change in $f$ and $dx$ is a small change in $x$.
"With respect to what" matters
Taking derivatives of the same function with respect to different variables will produce different results. In physics, these different results might have different dimensions and thus the derivative can have wildly different physical interpretations based off of "'with respect to' what".
Representations used
There is a partial derivative in every direction at any point
Derivatives can be found while holding one or more variables constant
For a smooth, two dimensional function, a partial derivative can be found at any point and in any direction.