How to Memorize Derivatives for Calc (Rules + Trig)
The fastest route for how to memorize derivatives is to sort them into tiers, learn the patterns that pair them up (so you memorize half as many), then drill them on flashcards and mix them into real problems until you stop thinking about them. You need about seven rules and around 16 specific derivatives for a typical Calc 1 or AP Calc course, and most of the trig ones come in matched pairs.
If you keep blanking on which trig derivatives get the negative sign, FlashDeck turns your derivative list into cards and resurfaces exactly the ones you keep missing: Build my study plan.
In the UK this is "differentiation" (A-level Maths and Further Maths); in Canada it's usually Grade 12 Calculus and Vectors or first-year calc. Same derivatives, different names.
The derivative rules you must know
These are the most important derivative rules. Every problem uses at least one of them.
| Rule | Formula | Example |
|---|---|---|
| Constant | d/dx [c] = 0 | d/dx [7] = 0 |
| Power | d/dx [xⁿ] = n·xⁿ⁻¹ | d/dx [x⁵] = 5x⁴ |
| Constant multiple | d/dx [c·f] = c·f′ | d/dx [3x²] = 6x |
| Sum/difference | (f ± g)′ = f′ ± g′ | d/dx [x³ + x] = 3x² + 1 |
| Product | (fg)′ = f′g + fg′ | d/dx [x·sin x] = sin x + x cos x |
| Quotient | (f/g)′ = (f′g − fg′) / g² | see below |
| Chain | d/dx [f(g(x))] = f′(g(x))·g′(x) | d/dx [(3x+1)⁴] = 12(3x+1)³ |
The quotient rule song
"Low d-high minus high d-low, over the square of what's below." Low is the bottom function, high is the top, d means "derivative of." It's the classic mnemonic, and it's in every calc class for a reason: the order matters and the minus sign makes it easy to flip.
Chain rule in one sentence
Derivative of the outside (leave the inside alone) times derivative of the inside. When you see a function inside another function, the chain rule is coming.
Patterns that cut memorizing in half
Exponentials and logs
- d/dx [eˣ] = eˣ (the function that's its own derivative)
- d/dx [aˣ] = aˣ · ln a
- d/dx [ln x] = 1/x
- d/dx [logₐ x] = 1 / (x · ln a)
Pattern: the base-a versions are just the e versions with a ln a stuck on. For aˣ it multiplies; for logₐ x it goes in the denominator. Learn eˣ and ln x, then add the ln a.
Understanding beats memorizing
If you're asking how to understand derivatives rather than just memorize them: a derivative is the slope (rate of change) of a function at a point. That's why the derivative of a constant is 0 (flat line, zero slope) and why sin x's derivative is cos x (sin is steepest where cos is biggest, at x = 0). Knowing the "why" gives you a way to sanity-check your memory.
Trig derivatives and the co- trick
Here are all six:
| Function | Derivative |
|---|---|
| sin x | cos x |
| cos x | −sin x |
| tan x | sec²x |
| cot x | −csc²x |
| sec x | sec x tan x |
| csc x | −csc x cot x |
The co- trick
Notice the pairs: sin/cos, tan/cot, sec/csc. Every function starting with "co" has a negative derivative. And each co-function's derivative is the non-co version with every function swapped for its co-partner:
- tan → sec², so cot → −cosec² (−csc²)
- sec → sec tan, so cosec → −cosec cot (−csc cot)
So you really only memorize three: sin → cos, tan → sec², sec → sec tan. The other three come free with a minus sign and co-swaps. That's the whole trick for how to memorize trig derivatives.
(Unit circle shaky? Evaluating these at specific angles needs it: how to memorize the unit circle.)
Inverse trig derivatives
These are the ones everyone dreads. Same co- trick, plus a few shapes.
| Function | Derivative |
|---|---|
| arcsin x | 1 / √(1 − x²) |
| arccos x | −1 / √(1 − x²) |
| arctan x | 1 / (1 + x²) |
| arccot x | −1 / (1 + x²) |
| arcsec x | 1 / (|x|√(x² − 1)) |
| arccsc x | −1 / (|x|√(x² − 1)) |
How to remember inverse trig derivatives
- Again, only three to learn. arcsin, arctan, arcsec. The co-versions are the same with a minus.
- arcsin has 1 − x² under a root. Think "sin is bounded, so the inside must stay positive for |x| < 1."
- arctan has 1 + x², no root. The friendliest one. It also shows up a lot in integration.
- arcsec has x² − 1 under the root, with |x| out front. Flipped order from arcsin.
Most courses emphasize arcsin and arctan. Check your syllabus before stressing about arcsec and arccsc.
Drill them with cards
Knowing the patterns is great; being instant on a timed test is the goal. That takes repetition spread over days, not one cram session.
Cards that work:
- Q: d/dx [csc x]? A: −csc x cot x
- Q: d/dx [arctan x]? A: 1 / (1 + x²)
- Q: Quotient rule? A: (f′g − fg′) / g²
- Q: d/dx [5ˣ]? A: 5ˣ ln 5
- Q: Which trig derivatives are negative? A: The co- ones: cos, cot, csc
Figuring out which derivatives you actually know and which you're just guessing on is the hard part. Paste your derivative list into FlashDeck and it makes one card per rule. Tap Knew it or Missed it; misses come back tomorrow, and the ones you know space out to 3 days, a week, then longer: Build my study plan.
A simple schedule
- Week 1: basic rules + power, exp and log derivatives. Five minutes of cards a day.
- Week 2: add the six trig derivatives.
- Week 3: add inverse trig.
- Every week after: keep reviewing only what's due.
Mix in practice problems
Cards make you fast at recall. Problems make you fast at choosing and combining. You need both.
Each day, do three to five problems that combine rules:
- d/dx [x² · eˣ] → product rule
- d/dx [sin(3x²)] → chain rule with trig: cos(3x²) · 6x
- d/dx [ln(tan x)] → chain rule with log and trig: sec²x / tan x
- d/dx [arctan(2x)] → chain rule with inverse trig: 2 / (1 + 4x²)
- d/dx [(x³ + 1) / cos x] → quotient rule
Mixing them (not doing 20 chain rule problems in a row) forces you to pick the rule each time, which is exactly what a test asks. Once derivatives feel easy, integrals are the same list in reverse, see how to memorize integration formulas.
Make every derivative instant before the test
Paste your notes into FlashDeck and in seconds you get one card per rule and derivative, using only what you pasted. About ten minutes a day, it shows just the ones you're about to forget, so the rules are automatic and your test time goes to the actual problem.
FAQ
What are the basic rules of derivatives?
The constant, power, constant multiple, sum/difference, product, quotient and chain rules. With those plus a short list of specific derivatives (exponential, log and trig), you can differentiate almost anything in a first calc course.
How do I memorize derivatives of trig functions?
Learn three: sin → cos, tan → sec², sec → sec tan. Every co-function (cos, cot, csc) is the matching one with a minus sign and co-swaps, so cos → −sin, cot → −csc², csc → −csc cot.
How do I remember inverse trig derivatives?
Learn arcsin (1/√(1 − x²)), arctan (1/(1 + x²)) and arcsec (1/(|x|√(x² − 1))). The arccos, arccot and arccsc derivatives are the same with a negative sign.
What's the best way to learn derivatives for AP Calc?
Understand what a derivative means, memorize the rules with flashcards spread over a few weeks, and do mixed practice problems daily. Spaced review keeps them fresh for the AP exam in May instead of just the unit test.
How long does it take to memorize derivatives?
Many students get the core list solid within two to three weeks of a few minutes of daily review, adding one group at a time. Cramming them in one night often works for a quiz but fades fast.