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 $x$.
You can flip a derivative
In other words, $\frac{dy}{dx} = \frac{1}{\frac{dx}{dy}}$. Note that this concept is challenging to express in Newton's notation, but arises naturally if implicit differentiation is covered.
Representations used
You can flip a partial derivative if the same variable(s) are constant
$\left(\frac{\partial f}{\partial x}\right)_y$
In other words,[\left(\frac{\partial y}{\partial z}\right)_z = \frac1{\left(\frac{\partial x}{\partial y}\right)_z}] Note that this equality is only true if the same variables are being held fixed on each side of the equality. This therefore relies on the thermodynamics notation that specifies what quantities are being held fixed for a partial derivative.