The Essential Calculus Formulas Every Student Needs to Memorize

One of the biggest struggles I see with calculus students — whether they're in AP Calc or college Calc 1 — is not having a clean, organized reference for the formulas they need to know cold.

The formulas themselves aren't complicated. But there are a lot of them, and mixing up even one sign or one trig derivative in the middle of a problem can tank your answer.

Here's the complete list, organized the way I teach it to my students. And if you want the printable version, I've got a free derivatives and integrals formula sheet you can download and keep at your desk.

<!-- IMAGE: A clean, organized formula sheet preview showing derivative rules grouped by category -->

Derivative Rules — The Foundation

These are the rules you'll use on virtually every problem. They need to be automatic.

Power Rule: $\frac{d}{dx}[x^n] = nx^{n-1}$

Constant Multiple Rule: $\frac{d}{dx}[cf(x)] = c \cdot f'(x)$

Sum/Difference Rule: $\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$

Product Rule:

<div class="latex-block"> \frac{d}{dx}[f(x) \cdot g(x)] = f'(x) \cdot g(x) + f(x) \cdot g'(x) </div>

Quotient Rule:

<div class="latex-block"> \frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{[g(x)]^2} </div>

Chain Rule:

<div class="latex-block"> \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) </div>

The chain rule is the one that gives students the most trouble, and it shows up in almost every problem from the second month of calculus onward. If you're struggling with it, check out the chain rule notes and practice quiz.

Trig Derivatives

You need all six of these memorized. No shortcuts, no looking them up mid-problem.

$\frac{d}{dx}[\sin x] = \cos x$

$\frac{d}{dx}[\cos x] = -\sin x$

$\frac{d}{dx}[\tan x] = \sec^2 x$

$\frac{d}{dx}[\cot x] = -\csc^2 x$

$\frac{d}{dx}[\sec x] = \sec x \tan x$

$\frac{d}{dx}[\csc x] = -\csc x \cot x$

Notice the pattern: the "co-" functions (cos, cot, csc) all have negative derivatives. That's a handy way to remember them.

Exponential and Logarithmic Derivatives

$\frac{d}{dx}[e^x] = e^x$

$\frac{d}{dx}[a^x] = a^x \ln a$

$\frac{d}{dx}[\ln x] = \frac{1}{x}$

$\frac{d}{dx}[\log_a x] = \frac{1}{x \ln a}$

The derivative of $e^x$ being itself is one of the most elegant facts in mathematics. It's also tested constantly on the AP exam.

Inverse Trig Derivatives

These come up more often than students expect, especially in integration:

$\frac{d}{dx}[\arcsin x] = \frac{1}{\sqrt{1-x^2}}$

$\frac{d}{dx}[\arccos x] = \frac{-1}{\sqrt{1-x^2}}$

$\frac{d}{dx}[\arctan x] = \frac{1}{1+x^2}$

<!-- IMAGE: A visual chart showing all derivative rules organized by category — basic rules, trig, exponential, inverse trig — color-coded for easy reference -->

Essential Integration Formulas

Integration is where having formulas memorized really pays off. You can't "figure out" most of these in the moment — you either know them or you don't.

Basic antiderivatives:

$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$

$\int \frac{1}{x} \, dx = \ln|x| + C$

$\int e^x \, dx = e^x + C$

$\int a^x \, dx = \frac{a^x}{\ln a} + C$

Trig integrals:

$\int \sin x \, dx = -\cos x + C$

$\int \cos x \, dx = \sin x + C$

$\int \sec^2 x \, dx = \tan x + C$

$\int \csc^2 x \, dx = -\cot x + C$

$\int \sec x \tan x \, dx = \sec x + C$

$\int \csc x \cot x \, dx = -\csc x + C$

Inverse trig integrals (these are critical for Calc 2):

$\int \frac{1}{\sqrt{1-x^2}} \, dx = \arcsin x + C$

$\int \frac{1}{1+x^2} \, dx = \arctan x + C$

Key Trig Identities for Calculus

You don't need every trig identity from precalculus, but you absolutely need these:

Pythagorean Identities:

$\sin^2 x + \cos^2 x = 1$

$1 + \tan^2 x = \sec^2 x$

$1 + \cot^2 x = \csc^2 x$

Double-Angle / Half-Angle (crucial for integration):

$\sin^2 x = \frac{1 - \cos 2x}{2}$

$\cos^2 x = \frac{1 + \cos 2x}{2}$

These last two show up constantly when you're integrating even powers of sine and cosine. If you're heading into Calculus 2, tape these to your desk.

The Fundamental Theorem of Calculus

Two parts, both essential:

Part 1: If $F(x) = \int_a^x f(t) \, dt$, then $F'(x) = f(x)$

Part 2: $\int_a^b f(x) \, dx = F(b) - F(a)$, where $F$ is any antiderivative of $f$

Part 1 is tested conceptually on the AP exam more often than students expect. Part 2 is what you use every time you evaluate a definite integral.

How to Actually Memorize All of This

Staring at a formula sheet doesn't work. Here's what does:

Write them by hand. Every morning for two weeks, write out all the derivative rules from memory. Check yourself against the sheet. The physical act of writing builds neural pathways that reading doesn't.

Use flashcards — but backwards. Instead of "what's the derivative of sin(x)?" try "which function has the derivative cos(x)?" Training both directions builds the kind of fluency you need for integration.

Practice problems, not formula lists. The formulas stick better when you use them in context. Work through problems using the Calculus 1 and Calculus 2 practice quizzes and reach for the formula sheet only when you're genuinely stuck.

Download and print. Grab the free derivatives and integrals formula sheet and keep it next to you every time you study. For Calc 2 students, the Calculus 2 formula sheet and sequences and series formula sheet cover the additional material you need.

<!-- IMAGE: A student writing formulas by hand in a notebook as a study technique, with a printed formula sheet nearby for reference -->

These formulas aren't hard individually. The challenge is having all of them accessible in your head at the same time, under exam pressure. That only comes from repetition and practice. Start now, review daily, and by exam time they'll be second nature.