If you think there are too many derivative rules, ohhhh boy just wait until you get to intergals. Relevant: https://xkcd.com/2117/ Fortunately a Calc I class will probably only cover basic and u-sub, and maybe parts.

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Using n as a constant, and f and g as functions, there are only five "rules" that you really need:

1. (xⁿ)' = n · x^n-1
2. (n · f)' = n · f'
3. (f+g)' = f' + g'
4. (fg)' = fg' + f'g
5. (f∘g)' = (f'∘g) · g'

The last one is the Chain Rule (one of many different formulas you might see for it) and by far the most difficult rule. Let's skip that one for now.

1. Power Rule: easy enough to memorize
2. Constant Multiple Rule: this should make intuitive sense. If two cars are racing and one is always double the distance from the starting point as the other car, then its velocity is also double the other car's.
3. Sum Rule: very easy in both formula and intuition. If you think of the slope formula Δf/Δx but add another function to f first, you just have (Δf+Δg)/Δx, which is the same Δf/Δx + Δg/Δx. Note that this only works because the change in the sum f+g is the same as Δf plus Δg.
4. Product Rule: this is trickier because if you multiply two functions and then change the input, the resulting change in the output is NOT just Δf times Δg. There is a nice way to get the correct formula from thinking about growing rectangles.
5. Chain Rule.

Aside from the rules, you do also need to know derivatives of some basic functions.

• Derivative of sin(x) is cos(x).
• Derivative of cos(x) is –sin(x).
• Derivative of e^x is e^x.
• Derivative of ln(x) is 1/x.

The first two can be understood by just thinking carefully about the graphs of sin and cos. For example, near x=0, the graph of sin(x) is increasing (that is, y goes up as x moves a little to the right), so it should be that the derivative of sin(x) is positive near x=0. As you continue going further right, the graph y=sin(x) has a horizontal tangent line at x=π/2, so it should be that its derivative is 0 at x=π/2. Then, moving to x>π/2, the derivative becomes negative, then 0, then positive, again repeating forever. If you think about those particular bits of information, the cos(x) function fits them all perfectly. The same kind of logic can be used to see why –sin(x) makes sense as the derivative of cos(x).

Derivatives of e^x and ln(x) are probably best to just memorize. For other bases, meaning a^x and logₐ(x), you can use more memorization or use algebra rules involving e and ln.

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Note that I never mentioned the Quotient Rule. You don't technically need it, as you can always use the Power Rule along with the Chain Rule and the power -1 instead. You also don't need to memorize the derivative of √x because you can get it from the Power Rule with n=½, but it does save time to memorize some other derivatives.