CaLaun

Quick Reference

All the formulas you need in one spot. Tap any one to try it out.

Derivative Rules

Basic Rules

\(\frac{d}{dx}[c] = 0\)
Constant Rule
Try: d/dx[5] →
\(\frac{d}{dx}[x^n] = nx^{n-1}\)
Power Rule
Try: polynomial →
\(\frac{d}{dx}[cf(x)] = c \cdot f'(x)\)
Constant Multiple
Try: 5·sin(x) →
\(\frac{d}{dx}[f + g] = f' + g'\)
Sum Rule
Try: eˣ + ln(x) →
\(\frac{d}{dx}[fg] = f'g + fg'\)
Product Rule
Try: x²·eˣ →
\(\frac{d}{dx}\left[\frac{f}{g}\right] = \frac{f'g - fg'}{g^2}\)
Quotient Rule
Try: tan(x) →
\(\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\)
Chain Rule
Try: √(x²+1) →

Trigonometric Functions

\(\frac{d}{dx}[\sin x] = \cos x\)
Try: sin(2x) →
\(\frac{d}{dx}[\cos x] = -\sin x\)
Try: cos(3x) →
\(\frac{d}{dx}[\tan x] = \sec^2 x\)
Try: tan(3x) →
\(\frac{d}{dx}[\cot x] = -\csc^2 x\)
Try: x·cot(x) →
\(\frac{d}{dx}[\sec x] = \sec x \tan x\)
Try: sec²(x) →
\(\frac{d}{dx}[\csc x] = -\csc x \cot x\)
Try: csc(x)/x →

Inverse Trigonometric Functions

\(\frac{d}{dx}[\arcsin x] = \frac{1}{\sqrt{1-x^2}}\)
Try: arcsin(2x) →
\(\frac{d}{dx}[\arccos x] = \frac{-1}{\sqrt{1-x^2}}\)
Try: arccos(x/2) →
\(\frac{d}{dx}[\arctan x] = \frac{1}{1+x^2}\)
Try: arctan(x²) →

Exponential & Logarithmic

\(\frac{d}{dx}[e^x] = e^x\)
Try: e^(x²) →
\(\frac{d}{dx}[a^x] = a^x \ln a\)
Try: 2^(3x) →
\(\frac{d}{dx}[\ln x] = \frac{1}{x}\)
Try: ln(x²+1) →
\(\frac{d}{dx}[\log_a x] = \frac{1}{x \ln a}\)
Try: log₁₀(x²) →

Advanced Examples

\(\frac{d}{dx}[x^x]\)
Logarithmic Differentiation
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\(\frac{d}{dx}[e^{x}\sin(x)]\)
Product + Chain Rules
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\(\frac{d}{dx}\left[\frac{x^2 - 1}{x^2 + 1}\right]\)
Quotient Rule
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\(\frac{d}{dx}[\ln(\sin(x))]\)
Nested Chain Rule
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Integration Formulas

Basic Integrals

\(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\)
Power Rule (n ≠ -1)
Try: polynomial →
\(\int \frac{1}{x} \, dx = \ln|x| + C\)
Try: 1/(2x) →
\(\int e^x \, dx = e^x + C\)
Try: e^(2x) →
\(\int a^x \, dx = \frac{a^x}{\ln a} + C\)
Try: 3^x →

Trigonometric Integrals

\(\int \sin x \, dx = -\cos x + C\)
Try: sin(3x) →
\(\int \cos x \, dx = \sin x + C\)
Try: cos(2x) →
\(\int \sec^2 x \, dx = \tan x + C\)
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\(\int \csc^2 x \, dx = -\cot x + C\)
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\(\int \sin^2 x \, dx\)
Power-Reducing Formula
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\(\int \cos^2 x \, dx\)
Power-Reducing Formula
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\(\int \tan x \, dx = -\ln|\cos x| + C\)
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\(\int \sin x \cos x \, dx\)
Trig Identity
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Inverse Trig Forms

\(\int \frac{1}{\sqrt{1-x^2}} \, dx = \arcsin x + C\)
Try: 1/√(4-x²) →
\(\int \frac{1}{1+x^2} \, dx = \arctan x + C\)
Try: 1/(9+x²) →
\(\int \frac{1}{x\sqrt{x^2-1}} \, dx = \text{arcsec}|x| + C\)
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Integration Techniques

\(\int u \, dv = uv - \int v \, du\)
Integration by Parts
Try: x²·eˣ →
\(\int f(g(x))g'(x) \, dx = F(g(x)) + C\)
U-Substitution
Try: x·√(x²+1) →
\(\int \frac{P(x)}{Q(x)} \, dx\)
Partial Fractions
Try: 1/((x-1)(x+2)) →
\(\int \ln x \, dx\)
By Parts (Classic)
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Advanced Examples

\(\int x \ln(x) \, dx\)
By Parts
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\(\int e^x \sin(x) \, dx\)
By Parts (Twice)
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\(\int \frac{x^2}{x^2 + 1} \, dx\)
Long Division
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\(\int \sqrt{1 - x^2} \, dx\)
Trig Substitution
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Limit Formulas

Important Limits

\(\lim_{x \to 0} \frac{\sin x}{x} = 1\)
Fundamental Trig Limit
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\(\lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2}\)
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\(\lim_{x \to 0} \frac{e^x - 1}{x} = 1\)
Exponential Limit
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\(\lim_{x \to 0} \frac{\ln(1+x)}{x} = 1\)
Logarithmic Limit
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\(\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e\)
Definition of e
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\(\lim_{x \to 0} \frac{\tan x}{x} = 1\)
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Indeterminate Forms (0/0)

\(\lim_{x \to 2} \frac{x^2 - 4}{x - 2}\)
Factor & Cancel
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\(\lim_{x \to 1} \frac{x^3 - 1}{x - 1}\)
Factor Cubic
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\(\lim_{x \to 0} \frac{\sqrt{1+x} - 1}{x}\)
Conjugate Multiply
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\(\lim_{x \to 4} \frac{\sqrt{x} - 2}{x - 4}\)
Rationalize
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Limits at Infinity

\(\lim_{x \to \infty} \frac{1}{x^n} = 0\) for n > 0
Try: 1/x³ →
\(\lim_{x \to \infty} \frac{2x^2 + 3x}{x^2 - 1}\)
Rational Function
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\(\lim_{x \to \infty} e^{-x} = 0\)
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\(\lim_{x \to \infty} \frac{e^x}{x^n}\)
Exponential Dominates
Try: eˣ/x⁵ →
\(\lim_{x \to \infty} \frac{\ln x}{x} = 0\)
Log Growth
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\(\lim_{x \to \infty} x \cdot e^{-x} = 0\)
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One-Sided Limits

\(\lim_{x \to 0^+} \frac{1}{x} = +\infty\)
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\(\lim_{x \to 0^-} \frac{1}{x} = -\infty\)
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\(\lim_{x \to 0^+} \ln x = -\infty\)
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\(\lim_{x \to 0^+} x \ln x = 0\)
0·∞ Form
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L'Hôpital's Rule

\(\text{If } \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{0}{0} \text{ or } \frac{\pm\infty}{\pm\infty}, \text{ then } \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}\)
L'Hôpital's Rule for Indeterminate Forms
Try: (eˣ-1-x)/x² →