Calculus I Formula Reference Card
Essential formulas for Calculus I — derivatives, integrals, limits, and theorems.
Limit Laws
Given \(\lim_{x \to a} f(x) = L\) and \(\lim_{x \to a} g(x) = M\):
| Law | Formula |
|---|---|
Sum/Difference |
\(\lim_{x \to a} [f(x) \pm g(x)] = L \pm M\) |
Constant Multiple |
\(\lim_{x \to a} [c \cdot f(x)] = c \cdot L\) |
Product |
\(\lim_{x \to a} [f(x) \cdot g(x)] = L \cdot M\) |
Quotient |
\(\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{L}{M}, \quad M \neq 0\) |
Power |
\(\lim_{x \to a} [f(x)]^n = L^n\) |
Squeeze Theorem |
If \(g(x) \leq f(x) \leq h(x)\) and \(\lim g = \lim h = L\), then \(\lim f = L\) |
Special Limits
-
\(\lim_{x \to 0} \frac{\sin x}{x} = 1\)
-
\(\lim_{x \to 0} \frac{1 - \cos x}{x} = 0\)
-
\(\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e\)
-
\(\lim_{x \to 0} \frac{e^x - 1}{x} = 1\)
Derivative Rules
Basic Rules
| Rule | Formula |
|---|---|
Constant |
\(\frac{d}{dx}[c] = 0\) |
Power Rule |
\(\frac{d}{dx}[x^n] = n x^{n-1}\) |
Constant Multiple |
\(\frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x)\) |
Sum/Difference |
\(\frac{d}{dx}[f \pm g] = f' \pm g'\) |
Product Rule |
\(\frac{d}{dx}[f \cdot g] = f' g + f g'\) |
Quotient Rule |
\(\frac{d}{dx}\left[\frac{f}{g}\right] = \frac{f' g - f g'}{g^2}\) |
Chain Rule |
\(\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\) |
Trigonometric Derivatives
| Function | Derivative |
|---|---|
\(\sin x\) |
\(\cos x\) |
\(\cos x\) |
\(-\sin x\) |
\(\tan x\) |
\(\sec^2 x\) |
\(\cot x\) |
\(-\csc^2 x\) |
\(\sec x\) |
\(\sec x \tan x\) |
\(\csc x\) |
\(-\csc x \cot x\) |
Inverse Trigonometric Derivatives
| Function | Derivative |
|---|---|
\(\arcsin x\) |
\(\frac{1}{\sqrt{1 - x^2}}\) |
\(\arccos x\) |
\(-\frac{1}{\sqrt{1 - x^2}}\) |
\(\arctan x\) |
\(\frac{1}{1 + x^2}\) |
Exponential and Logarithmic Derivatives
| Function | Derivative |
|---|---|
\(e^x\) |
\(e^x\) |
\(a^x\) |
\(a^x \ln a\) |
\(\ln x\) |
\(\frac{1}{x}\) |
\(\log_a x\) |
\(\frac{1}{x \ln a}\) |
Integration Formulas
Basic Integrals
| Integral | Result |
|---|---|
\(\int x^n \, dx\) |
\(\frac{x^{n+1}}{n+1} + C, \quad n \neq -1\) |
\(\int \frac{1}{x} \, dx\) |
latexmath:[\ln |
x |
+ C] |
\(\int e^x \, dx\) |
\(e^x + C\) |
\(\int a^x \, dx\) |
\(\frac{a^x}{\ln a} + C\) |
Trigonometric Integrals
| Integral | Result |
|---|---|
\(\int \sin x \, dx\) |
\(-\cos x + C\) |
\(\int \cos x \, dx\) |
\(\sin x + C\) |
\(\int \sec^2 x \, dx\) |
\(\tan x + C\) |
\(\int \csc^2 x \, dx\) |
\(-\cot x + C\) |
\(\int \sec x \tan x \, dx\) |
\(\sec x + C\) |
\(\int \csc x \cot x \, dx\) |
\(-\csc x + C\) |
Integrals Producing Inverse Trig
| Integral | Result |
|---|---|
\(\int \frac{1}{\sqrt{1-x^2}} \, dx\) |
\(\arcsin x + C\) |
\(\int \frac{1}{1+x^2} \, dx\) |
\(\arctan x + C\) |
\(\int \frac{1}{x\sqrt{x^2-1}} \, dx\) |
\(\operatorname{arcsec} \lvert x \rvert + C\) |
Key Theorems
Intermediate Value Theorem (IVT)
If \(f\) is continuous on \(\lbrack a, b \rbrack\) and \(N\) is between \(f(a)\) and \(f(b)\), then there exists \(c \in (a, b)\) such that \(f(c) = N\).
Extreme Value Theorem (EVT)
If \(f\) is continuous on \(\lbrack a, b \rbrack\), then \(f\) attains an absolute maximum and minimum on \(\lbrack a, b \rbrack\).
Mean Value Theorem (MVT)
If \(f\) is continuous on \(\lbrack a, b \rbrack\) and differentiable on \((a, b)\), then there exists \(c \in (a, b)\) such that:
Rolle’s Theorem
Special case of MVT: if \(f(a) = f(b)\), then there exists \(c \in (a, b)\) such that \(f'(c) = 0\).
Fundamental Theorem of Calculus
Part 1: If \(f\) is continuous on \(\lbrack a, b \rbrack\), then \(F(x) = \int_a^x f(t) \, dt\) is differentiable and:
Part 2: If \(f\) is continuous on \(\lbrack a, b \rbrack\) and \(F\) is any antiderivative of \(f\), then:
Net Change Theorem
The integral of a rate of change gives the net change.