Single-Variable Calculus
The Derivative
The Product Rule
Being able to differentiate a single function, or a sum of them, is very useful, but we also need a way of differentiating products of functions to be able to work fully with them.
The Chain Rule
As you rise through the levels of a physics education, the chain rule, and being able to do it quickly in your head, becomes paramount. If you have ever found yourself staring at a bit of physics in some textbook, and scratching your head over how the author seemed to jump from one line to the next, it is likely the chain rule is involved.
The Quotient Rule
Do not use the quotient rule. I can genuinely not think of a single situation where it would be necessary to use the quotient rule. If you, dear reader, can, then I would love to hear what it is, and I will edit this section appropriately. Instead, you can simply use the product rule after recalling that any quotient can be rewritten as an inverse power of suitable value, for example,
\[\begin{align} \frac{1}{x} &= x^{-1} \\ \implies \dv{}{x}\left(\frac{1}{x}\right) &= -x^{-2} \end{align}\]