Pre Calc 2 Formulas

Absolutely! Here’s a complete Calculus 2 “Pre-Calc 2 Prep” Cheat Sheet combining everything we’ve discussed in this chat. I’ve organized it for clarity and memorization.

Calculus 2 Prep Cheat Sheet

1. Derivatives

Basic Rules:

ddx[xn]=nxn−1\frac{d}{dx}[x^n] = n x^{n-1}ddx[f(x)±g(x)]=f′(x)±g′(x)\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)ddx[f(x)g(x)]=f′g+fg′\frac{d}{dx}[f(x)g(x)] = f'g + fg'ddx[fg]=f′g−fg′g2\frac{d}{dx}\left[\frac{f}{g}\right] = \frac{f'g - fg'}{g^2}ddx[f(g(x))]=f′(g(x))⋅g′(x)\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)

Common Functions:

ddx[sin⁡x]=cos⁡x,ddx[cos⁡x]=−sin⁡x\frac{d}{dx}[\sin x] = \cos x, \quad \frac{d}{dx}[\cos x] = -\sin xddx[tan⁡x]=sec⁡2x,ddx[cot⁡x]=−csc⁡2x\frac{d}{dx}[\tan x] = \sec^2 x, \quad \frac{d}{dx}[\cot x] = -\csc^2 xddx[sec⁡x]=sec⁡xtan⁡x,ddx[csc⁡x]=−csc⁡xcot⁡x\frac{d}{dx}[\sec x] = \sec x \tan x, \quad \frac{d}{dx}[\csc x] = -\csc x \cot xddx[ex]=ex,ddx[ax]=axln⁡a\frac{d}{dx}[e^x] = e^x, \quad \frac{d}{dx}[a^x] = a^x \ln addx[ln⁡x]=1x,ddx[log⁡ax]=1xln⁡a\frac{d}{dx}[\ln x] = \frac{1}{x}, \quad \frac{d}{dx}[\log_a x] = \frac{1}{x \ln a}

2. Integrals (Antiderivatives)

Power Rule:

∫xn dx=xn+1n+1+C(n≠−1)\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)

Trig Integrals:

∫sin⁡x dx=−cos⁡x+C,∫cos⁡x dx=sin⁡x+C\int \sin x \, dx = -\cos x + C, \quad \int \cos x \, dx = \sin x + C∫tan⁡x dx=ln⁡∣sec⁡x∣+C,∫cot⁡x dx=ln⁡∣sin⁡x∣+C\int \tan x \, dx = \ln|\sec x| + C, \quad \int \cot x \, dx = \ln|\sin x| + C∫sec⁡x dx=ln⁡∣sec⁡x+tan⁡x∣+C,∫csc⁡x dx=ln⁡∣csc⁡x−cot⁡x∣+C\int \sec x \, dx = \ln|\sec x + \tan x| + C, \quad \int \csc x \, dx = \ln|\csc x - \cot x| + C

Exponential/Log:

∫ex dx=ex+C,∫ax dx=axln⁡a+C\int e^x \, dx = e^x + C, \quad \int a^x \, dx = \frac{a^x}{\ln a} + C∫1x dx=ln⁡∣x∣+C\int \frac{1}{x} \, dx = \ln|x| + C

Sum/Difference Rule:

∫[f(x)±g(x)]dx=∫f(x)dx±∫g(x)dx\int [f(x) \pm g(x)] dx = \int f(x) dx \pm \int g(x) dx

3. Trig Functions & Identities

Basic Functions & Reciprocals:

Pythagorean Identities:

sin⁡2x+cos⁡2x=1\sin^2 x + \cos^2 x = 11+tan⁡2x=sec⁡2x1 + \tan^2 x = \sec^2 x1+cot⁡2x=csc⁡2x1 + \cot^2 x = \csc^2 x

Double-Angle:

sin⁡2x=2sin⁡xcos⁡x\sin 2x = 2 \sin x \cos xcos⁡2x=cos⁡2x−sin⁡2x=2cos⁡2x−1=1−2sin⁡2x\cos 2x = \cos^2 x - \sin^2 x = 2\cos^2 x - 1 = 1 - 2\sin^2 x

Half-Angle:

sin⁡2x=1−cos⁡2x2,cos⁡2x=1+cos⁡2x2\sin^2 x = \frac{1 - \cos 2x}{2}, \quad \cos^2 x = \frac{1 + \cos 2x}{2}

4. Algebra & Logs

Factorization:

a2−b2=(a−b)(a+b)a^2 - b^2 = (a-b)(a+b)a3±b3=(a±b)(a2∓ab+b2)a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)

Log Rules:

ln⁡(ab)=ln⁡a+ln⁡b,ln⁡(a/b)=ln⁡a−ln⁡b,ln⁡(ab)=bln⁡a\ln(ab) = \ln a + \ln b, \quad \ln(a/b) = \ln a - \ln b, \quad \ln(a^b) = b \ln a

5. Limits & Series Foundations

Important Limits:

lim⁡x→0sin⁡xx=1,lim⁡x→∞1x=0\lim_{x \to 0} \frac{\sin x}{x} = 1, \quad \lim_{x \to \infty} \frac{1}{x} = 0

Series:

1+r+r2+⋯+rn−1=1−rn1−r,∣r∣<11 + r + r^2 + \dots + r^{n-1} = \frac{1-r^n}{1-r}, \quad |r|<1

L’Hospital’s Rule:
For 0/0 or ∞/∞:

lim⁡x→af(x)g(x)=lim⁡x→af′(x)g′(x)\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}

This is basically everything you should already know heading into Calc 2: derivatives, integrals, trig functions, identities, logs, algebra, limits, and basic series.

If you want, I can also make a Condensed “1-page Calc 2 Formulas” sheet that’s visually organized with colors for derivatives, integrals, and trig so you can memorize it super fast.

Do you want me to make that version too?