Calculus Formula Handbook

Differentiation & Integration at Your Fingertips

Table of Contents

1. Limits

Basic Limit Properties

\[ \begin{aligned} \lim_{x \to a} [f(x) \pm g(x)] &= \lim_{x \to a} f(x) \pm \lim_{x \to a} g(x) \\[10pt] \lim_{x \to a} [f(x) \cdot g(x)] &= \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x) \\[10pt] \lim_{x \to a} \frac{f(x)}{g(x)} &= \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}, \quad \text{if } \lim_{x \to a} g(x) \neq 0 \\[10pt] \lim_{x \to a} [cf(x)] &= c \lim_{x \to a} f(x) \\[10pt] \lim_{x \to a} [f(x)]^n &= \left[\lim_{x \to a} f(x)\right]^n \end{aligned} \]

Important Limit Formulas

\[ \begin{aligned} \lim_{x \to 0} \frac{\sin x}{x} &= 1 \\[10pt] \lim_{x \to 0} \frac{\tan x}{x} &= 1 \\[10pt] \lim_{x \to 0} \frac{1 - \cos x}{x} &= 0 \\[10pt] \lim_{x \to 0} \frac{1 - \cos x}{x^2} &= \frac{1}{2} \\[10pt] \lim_{x \to 0} (1 + x)^{1/x} &= e \\[10pt] \lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x &= e \\[10pt] \lim_{x \to 0} \frac{e^x - 1}{x} &= 1 \\[10pt] \lim_{x \to 0} \frac{\ln(1 + x)}{x} &= 1 \\[10pt] \lim_{x \to 0} \frac{a^x - 1}{x} &= \ln a \end{aligned} \]
L'Hôpital's Rule: If \(\lim_{x \to a} \frac{f(x)}{g(x)}\) gives \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), then: \[ \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} \]

2. Derivatives

Basic Derivatives

Function Derivative
\(k\) (constant) \(\frac{d}{dx}(k) = 0\)
\(x\) \(\frac{d}{dx}(x) = 1\)
\(x^n\) \(\frac{d}{dx}(x^n) = nx^{n-1}\)
\(\sqrt{x}\) \(\frac{d}{dx}(\sqrt{x}) = \frac{1}{2\sqrt{x}}\)
\(\frac{1}{x}\) \(\frac{d}{dx}\left(\frac{1}{x}\right) = -\frac{1}{x^2}\)

Exponential & Logarithmic Derivatives

Function Derivative
\(e^x\) \(\frac{d}{dx}(e^x) = e^x\)
\(e^{kx}\) \(\frac{d}{dx}(e^{kx}) = ke^{kx}\)
\(a^x\) \(\frac{d}{dx}(a^x) = a^x \ln a\)
\(\ln x\) \(\frac{d}{dx}(\ln x) = \frac{1}{x}\)
\(\ln |x|\) \(\frac{d}{dx}(\ln |x|) = \frac{1}{x}\)
\(\log_a x\) \(\frac{d}{dx}(\log_a x) = \frac{1}{x \ln a}\)

Trigonometric Derivatives

Function Derivative
\(\sin x\) \(\frac{d}{dx}(\sin x) = \cos x\)
\(\cos x\) \(\frac{d}{dx}(\cos x) = -\sin x\)
\(\tan x\) \(\frac{d}{dx}(\tan x) = \sec^2 x\)
\(\cot x\) \(\frac{d}{dx}(\cot x) = -\csc^2 x\)
\(\sec x\) \(\frac{d}{dx}(\sec x) = \sec x \tan x\)
\(\csc x\) \(\frac{d}{dx}(\csc x) = -\csc x \cot x\)

Inverse Trigonometric Derivatives

Function Derivative
\(\sin^{-1} x\) \(\frac{d}{dx}(\sin^{-1} x) = \frac{1}{\sqrt{1 - x^2}}\)
\(\cos^{-1} x\) \(\frac{d}{dx}(\cos^{-1} x) = -\frac{1}{\sqrt{1 - x^2}}\)
\(\tan^{-1} x\) \(\frac{d}{dx}(\tan^{-1} x) = \frac{1}{1 + x^2}\)
\(\cot^{-1} x\) \(\frac{d}{dx}(\cot^{-1} x) = -\frac{1}{1 + x^2}\)
\(\sec^{-1} x\) \(\frac{d}{dx}(\sec^{-1} x) = \frac{1}{|x|\sqrt{x^2 - 1}}\)
\(\csc^{-1} x\) \(\frac{d}{dx}(\csc^{-1} x) = -\frac{1}{|x|\sqrt{x^2 - 1}}\)

Hyperbolic Derivatives

Function Derivative
\(\sinh x\) \(\frac{d}{dx}(\sinh x) = \cosh x\)
\(\cosh x\) \(\frac{d}{dx}(\cosh x) = \sinh x\)
\(\tanh x\) \(\frac{d}{dx}(\tanh x) = \text{sech}^2 x\)
\(\text{coth} x\) \(\frac{d}{dx}(\text{coth} x) = -\text{csch}^2 x\)
\(\text{sech} x\) \(\frac{d}{dx}(\text{sech} x) = -\text{sech} x \tanh x\)
\(\text{csch} x\) \(\frac{d}{dx}(\text{csch} x) = -\text{csch} x \text{coth} x\)

Inverse Hyperbolic Derivatives

Function Derivative
\(\sinh^{-1} x\) \(\frac{d}{dx}(\sinh^{-1} x) = \frac{1}{\sqrt{x^2 + 1}}\)
\(\cosh^{-1} x\) \(\frac{d}{dx}(\cosh^{-1} x) = \frac{1}{\sqrt{x^2 - 1}}\)
\(\tanh^{-1} x\) \(\frac{d}{dx}(\tanh^{-1} x) = \frac{1}{1 - x^2}\)

3. Important Derivative Rules

Sum & Difference Rule

\[ \frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x) \]

Product Rule

\[ \frac{d}{dx}[f(x) \cdot g(x)] = f'(x)g(x) + f(x)g'(x) \]

Quotient Rule

\[ \frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2} \]

Chain Rule

\[ \frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \]

Or equivalently: \(\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\)

Power Rule (General)

\[ \frac{d}{dx}[f(x)]^n = n[f(x)]^{n-1} \cdot f'(x) \]

Implicit Differentiation

For equations where y is not explicitly solved: differentiate both sides with respect to x, treating y as a function of x.

Logarithmic Differentiation

For \(y = f(x)\): Take \(\ln\) of both sides, then differentiate:

\[ \frac{1}{y}\frac{dy}{dx} = \frac{d}{dx}[\ln f(x)] \quad \Rightarrow \quad \frac{dy}{dx} = y \cdot \frac{d}{dx}[\ln f(x)] \]

Higher Order Derivatives

\[ \begin{aligned} f'(x) \text{ or } \frac{dy}{dx} &\quad \text{(First derivative)} \\[8pt] f''(x) \text{ or } \frac{d^2y}{dx^2} &\quad \text{(Second derivative)} \\[8pt] f^{(n)}(x) \text{ or } \frac{d^ny}{dx^n} &\quad \text{(nth derivative)} \end{aligned} \]

4. Integrals

Basic Integrals

Function Integral
\(\int k \, dx\) \(kx + C\)
\(\int x^n \, dx\) \(\frac{x^{n+1}}{n+1} + C, \quad n \neq -1\)
\(\int \frac{1}{x} \, dx\) \(\ln |x| + C\)
\(\int \frac{1}{x^2} \, dx\) \(-\frac{1}{x} + C\)
\(\int \sqrt{x} \, dx\) \(\frac{2}{3}x^{3/2} + C\)
\(\int \frac{1}{\sqrt{x}} \, dx\) \(2\sqrt{x} + C\)

Exponential & Logarithmic Integrals

Function Integral
\(\int e^x \, dx\) \(e^x + C\)
\(\int e^{kx} \, dx\) \(\frac{1}{k}e^{kx} + C\)
\(\int a^x \, dx\) \(\frac{a^x}{\ln a} + C\)
\(\int \ln x \, dx\) \(x\ln x - x + C\)
\(\int \frac{\ln x}{x} \, dx\) \(\frac{(\ln x)^2}{2} + C\)

Trigonometric Integrals

Function Integral
\(\int \sin x \, dx\) \(-\cos x + C\)
\(\int \cos x \, dx\) \(\sin x + C\)
\(\int \tan x \, dx\) \(\ln |\sec x| + C = -\ln |\cos x| + C\)
\(\int \cot x \, dx\) \(\ln |\sin x| + C\)
\(\int \sec x \, dx\) \(\ln |\sec x + \tan x| + C\)
\(\int \csc x \, dx\) \(-\ln |\csc x + \cot x| + C\)
\(\int \sec^2 x \, dx\) \(\tan x + C\)
\(\int \csc^2 x \, dx\) \(-\cot x + C\)
\(\int \sec x \tan x \, dx\) \(\sec x + C\)
\(\int \csc x \cot x \, dx\) \(-\csc x + C\)
\(\int \sin^2 x \, dx\) \(\frac{x}{2} - \frac{\sin 2x}{4} + C\)
\(\int \cos^2 x \, dx\) \(\frac{x}{2} + \frac{\sin 2x}{4} + C\)

Inverse Trigonometric Integrals

Function Integral
\(\int \frac{dx}{\sqrt{a^2 - x^2}}\) \(\sin^{-1}\left(\frac{x}{a}\right) + C\)
\(\int \frac{dx}{a^2 + x^2}\) \(\frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right) + C\)
\(\int \frac{dx}{x\sqrt{x^2 - a^2}}\) \(\frac{1}{a}\sec^{-1}\left(\frac{|x|}{a}\right) + C\)
\(\int \sin^{-1} x \, dx\) \(x\sin^{-1} x + \sqrt{1-x^2} + C\)
\(\int \cos^{-1} x \, dx\) \(x\cos^{-1} x - \sqrt{1-x^2} + C\)
\(\int \tan^{-1} x \, dx\) \(x\tan^{-1} x - \frac{1}{2}\ln(1+x^2) + C\)

Hyperbolic Integrals

Function Integral
\(\int \sinh x \, dx\) \(\cosh x + C\)
\(\int \cosh x \, dx\) \(\sinh x + C\)
\(\int \tanh x \, dx\) \(\ln(\cosh x) + C\)
\(\int \text{coth} x \, dx\) \(\ln |\sinh x| + C\)
\(\int \text{sech}^2 x \, dx\) \(\tanh x + C\)
\(\int \text{csch}^2 x \, dx\) \(-\text{coth} x + C\)

Special Integrals (Reduction Formulas)

Function Integral
\(\int x e^x \, dx\) \(e^x(x-1) + C\)
\(\int x \sin x \, dx\) \(-x\cos x + \sin x + C\)
\(\int x \cos x \, dx\) \(x\sin x + \cos x + C\)
\(\int e^{ax}\sin(bx) \, dx\) \(\frac{e^{ax}}{a^2+b^2}[a\sin(bx) - b\cos(bx)] + C\)
\(\int e^{ax}\cos(bx) \, dx\) \(\frac{e^{ax}}{a^2+b^2}[a\cos(bx) + b\sin(bx)] + C\)
\(\int \sqrt{a^2 - x^2} \, dx\) \(\frac{x}{2}\sqrt{a^2-x^2} + \frac{a^2}{2}\sin^{-1}\left(\frac{x}{a}\right) + C\)
\(\int \sqrt{x^2 + a^2} \, dx\) \(\frac{x}{2}\sqrt{x^2+a^2} + \frac{a^2}{2}\ln|x+\sqrt{x^2+a^2}| + C\)
\(\int \sqrt{x^2 - a^2} \, dx\) \(\frac{x}{2}\sqrt{x^2-a^2} - \frac{a^2}{2}\ln|x+\sqrt{x^2-a^2}| + C\)

5. Integration Techniques & Properties

Basic Properties

Linearity Property:
\[ \int [af(x) + bg(x)] \, dx = a\int f(x) \, dx + b\int g(x) \, dx \]
Constant Multiple Rule:
\[ \int k \cdot f(x) \, dx = k\int f(x) \, dx \]
Sum and Difference Rule:
\[ \int [f(x) \pm g(x)] \, dx = \int f(x) \, dx \pm \int g(x) \, dx \]

Integration Techniques

Substitution Method (u-substitution)

Let \(u = g(x)\), then \(du = g'(x)dx\)

\[ \int f(g(x)) \cdot g'(x) \, dx = \int f(u) \, du \]

Integration by Parts

\[ \int u \, dv = uv - \int v \, du \]

Choose u using LIATE rule: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential

Partial Fractions

Decompose rational functions:

\[ \frac{P(x)}{Q(x)} = \frac{A}{x-a} + \frac{B}{x-b} + \cdots \]

Trigonometric Substitution

Fundamental Theorem of Calculus (Part 1):
\[ \text{If } F(x) = \int_a^x f(t) \, dt, \text{ then } F'(x) = f(x) \]
Fundamental Theorem of Calculus (Part 2):
\[ \int_a^b f(x) \, dx = F(b) - F(a) \]
where \(F'(x) = f(x)\)

6. Definite Integrals

Properties of Definite Integrals

Zero Interval:
\[ \int_a^a f(x) \, dx = 0 \]
Reversal of Limits:
\[ \int_a^b f(x) \, dx = -\int_b^a f(x) \, dx \]
Additivity:
\[ \int_a^b f(x) \, dx + \int_b^c f(x) \, dx = \int_a^c f(x) \, dx \]
Constant Multiple:
\[ \int_a^b k \cdot f(x) \, dx = k\int_a^b f(x) \, dx \]
Integral of Constant:
\[ \int_a^b c \, dx = c(b - a) \]
Comparison Property:
\[ \text{If } f(x) \leq g(x) \text{ on } [a,b], \text{ then } \int_a^b f(x) \, dx \leq \int_a^b g(x) \, dx \]
Even Function Property:
\[ \text{If } f(-x) = f(x), \text{ then } \int_{-a}^a f(x) \, dx = 2\int_0^a f(x) \, dx \]
Odd Function Property:
\[ \text{If } f(-x) = -f(x), \text{ then } \int_{-a}^a f(x) \, dx = 0 \]
Absolute Value Rule:
\[ \left|\int_a^b f(x) \, dx\right| \leq \int_a^b |f(x)| \, dx \]

Applications of Definite Integrals

Area under a curve:
\[ A = \int_a^b f(x) \, dx \]
Area between two curves:
\[ A = \int_a^b [f(x) - g(x)] \, dx \quad \text{where } f(x) \geq g(x) \]
Volume of Revolution (Disk Method):
\[ V = \pi\int_a^b [f(x)]^2 \, dx \]
Volume of Revolution (Shell Method):
\[ V = 2\pi\int_a^b x \cdot f(x) \, dx \]
Arc Length:
\[ L = \int_a^b \sqrt{1 + [f'(x)]^2} \, dx \]
Surface Area of Revolution:
\[ S = 2\pi\int_a^b f(x)\sqrt{1 + [f'(x)]^2} \, dx \]

7. Series & Sequences

Important Series

Geometric Series:
\[ \sum_{n=0}^{\infty} ar^n = \frac{a}{1-r}, \quad |r| < 1 \]
Harmonic Series:
\[ \sum_{n=1}^{\infty} \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \cdots \quad \text{(Diverges)} \]
p-Series:
\[ \sum_{n=1}^{\infty} \frac{1}{n^p} \text{ converges if } p > 1, \text{ diverges if } p \leq 1 \]

Taylor Series

\[ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots \]

Maclaurin Series (Taylor series at a = 0)

Exponential Function:
\[ e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \]
Sine Function:
\[ \sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots \]
Cosine Function:
\[ \cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots \]
Natural Logarithm:
\[ \ln(1+x) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1} x^n}{n} = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots, \quad |x| < 1 \]
Binomial Series:
\[ (1+x)^n = \sum_{k=0}^{\infty} \binom{n}{k} x^k = 1 + nx + \frac{n(n-1)}{2!}x^2 + \cdots, \quad |x| < 1 \]

Convergence Tests

Ratio Test: For \(\sum a_n\), if \(\lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| = L\), then:
Root Test: For \(\sum a_n\), if \(\lim_{n \to \infty} \sqrt[n]{|a_n|} = L\), then:
Integral Test: If \(f(x)\) is positive, continuous, and decreasing, then \(\sum_{n=1}^{\infty} f(n)\) and \(\int_1^{\infty} f(x) \, dx\) either both converge or both diverge.
Comparison Test: If \(0 \leq a_n \leq b_n\):
Alternating Series Test: \(\sum (-1)^n a_n\) converges if:

Additional Important Formulas

Partial Derivatives

\[ \frac{\partial f}{\partial x} = \lim_{h \to 0} \frac{f(x+h, y) - f(x, y)}{h} \]

Gradient

\[ \nabla f = \left\langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right\rangle \]

Directional Derivative

\[ D_{\mathbf{u}}f = \nabla f \cdot \mathbf{u} \]

Double Integral

\[ \iint_R f(x,y) \, dA = \int_a^b \int_c^d f(x,y) \, dy \, dx \]

Triple Integral

\[ \iiint_V f(x,y,z) \, dV = \int_a^b \int_c^d \int_e^f f(x,y,z) \, dz \, dy \, dx \]

Green's Theorem

\[ \oint_C (P \, dx + Q \, dy) = \iint_D \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) dA \]

Stokes' Theorem

\[ \oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} \]

Divergence Theorem

\[ \iiint_V (\nabla \cdot \mathbf{F}) \, dV = \iint_S \mathbf{F} \cdot d\mathbf{S} \]

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