Mathematics: Analysis formulary

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Unit 1Sequences and Series

Each term adds a constant common difference $d$ to the previous one.

$a_1$ is the first term; $n$ is the position.

Sum of the first $n$ terms: average of the ends times the count.

The defining limit for Euler's number $e$ .

Generalises the Euler limit; the source of continuous growth.

The constant factor between consecutive terms of a geometric sequence.

Each term multiplies the previous one by the common ratio $q$ .

Sum of the first $n$ terms of a geometric sequence.

Converges only when $|q| < 1$ ; otherwise the sum diverges.

A minus sign flips the Euler limit to $1/e$ .

The root grows fast enough to flatten $n$ down to 1.

Any fixed positive base under an nth root tends to 1.

A fixed numerator over a growing denominator goes to zero.

Repeatedly multiplying by a number smaller than 1 collapses to zero.

Unit 2Functions and Inverse Functions

Cosine is sine shifted by a quarter turn; it starts at 1.

Reciprocal of sine; its value never lands inside $(-1, 1)$ .

Undefined where sine is zero; ranges over all reals.

Reciprocal of cosine; its value never lands inside $(-1, 1)$ .

Defined for every real number; output never leaves $[-1, 1]$ .

Undefined where cosine is zero; ranges over all reals.

The quarter-turn values; sine starts at 0 and peaks at $\tfrac{\pi}{2}$ .

Unit 3Differential Calculus

Write it as $x^{-n}$ and apply the power rule.

Seen in Exam formula sheet.

A constant does not change, so its rate of change is zero.

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A general base picks up a factor of $\ln a$ .

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Same as arcsin but negated.

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The arctan derivative, negated.

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Only defined for $|x| > 1$ , where the root is real.

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Restricted to $|x| > 1$ .

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Valid on $(-1, 1)$ , where the square root stays real.

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Defined for all reals; no square root to worry about.

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A plus sign under the root, so it is defined for all reals.

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Restricted to $|x| < 1$ .

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The minus sign is the easy one to drop on an exam.

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No minus sign here, unlike $\cos$ .

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The cotangent mirror of the tangent derivative, with a minus sign.

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Valid away from $x = 0$ , where $\sinh$ vanishes.

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The exponential is its own derivative.

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The natural log has the simplest derivative of the logarithms.

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A change-of-base factor $1/\ln a$ over the natural log.

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Differentiating cycles sin -> cos -> -sin -> -cos.

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Hyperbolic functions differentiate without the sign flips of trig.

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Two equivalent forms; pick whichever the next step needs.

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Note the minus in $1 - \tanh^2 x$ , versus $1 + \tan^2 x$ for trig.

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A special case of the power rule with $n = \tfrac{1}{2}$ .

Seen in Exam formula sheet.

Bring the exponent down, then subtract one from it.

Seen in Exam formula sheet.

Unit 4Integral Calculus

The absolute value covers negative $x$ as well.

Seen in Exam formula sheet.

A constant integrates to a straight line; never forget the $+C$ .

Seen in Exam formula sheet.

Dividing by $\ln a$ undoes the factor that differentiation adds.

Seen in Exam formula sheet.

Comes from $u = \sin x$ substitution; no minus sign this time.

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A linear inner function divides by its coefficient $a$ .

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The exponential integrates to itself.

Seen in Exam formula sheet.

Comes from $u = \cos x$ substitution.

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Add one to the exponent, then divide by the new exponent.

The case $a = -1$ is the logarithm, handled separately.

Seen in Exam formula sheet.

Tap a card to unfold the notes.

Unit 1Sequences and Series

Arithmetic sequencean=a1+(n−1) da_n = a_1 + (n - 1)\,dan​=a1​+(n−1)d

Arithmetic seriesSn=n2 (a1+an)S_n = \frac{n}{2}\,(a_1 + a_n)Sn​=2n​(a1​+an​)

Euler limitlim⁡n→∞(1+1n)n=e\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^{n} = en→∞lim​(1+n1​)n=e

Euler limit with xlim⁡n→∞(1+xn)n=ex\lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^{n} = e^{x}n→∞lim​(1+nx​)n=ex

Geometric common ratioq=an+1anq = \frac{a_{n+1}}{a_n}q=an​an+1​​

Geometric sequencean=a1⋅q n−1a_n = a_1 \cdot q^{\,n-1}an​=a1​⋅qn−1

Geometric series (finite)Sn=a1⋅q n−1q−1,q≠1S_n = a_1 \cdot \frac{q^{\,n} - 1}{q - 1}, \quad q \neq 1Sn​=a1​⋅q−1qn−1​,q=1

Geometric series (infinite)S=a11−q,∣q∣<1S = \frac{a_1}{1 - q}, \quad |q| < 1S=1−qa1​​,∣q∣<1

Inverse Euler limitlim⁡n→∞(1−1n)n=1e\lim_{n \to \infty} \left(1 - \frac{1}{n}\right)^{n} = \frac{1}{e}n→∞lim​(1−n1​)n=e1​

Limit of the nth root of nlim⁡n→∞nn=1\lim_{n \to \infty} \sqrt[n]{n} = 1n→∞lim​nn​=1

Limit of the nth root of xlim⁡n→∞xn=1,x>0\lim_{n \to \infty} \sqrt[n]{x} = 1, \quad x > 0n→∞lim​nx​=1,x>0

Limit of x / nlim⁡n→∞xn=0,x∈R\lim_{n \to \infty} \frac{x}{n} = 0, \quad x \in \mathbb{R}n→∞lim​nx​=0,x∈R

Limit of x^nlim⁡n→∞xn=0,∣x∣<1\lim_{n \to \infty} x^{n} = 0, \quad |x| < 1n→∞lim​xn=0,∣x∣<1

Unit 2Functions and Inverse Functions

Cosine: special valuescos⁡0=1,cos⁡π2=0,cos⁡π=−1,cos⁡3π2=0\cos 0 = 1, \quad \cos\tfrac{\pi}{2} = 0, \quad \cos\pi = -1, \quad \cos\tfrac{3\pi}{2} = 0cos0=1,cos2π​=0,cosπ=−1,cos23π​=0

Domain of cosecantcsc⁡: R∖{kπ}→(−∞,−1]∪[1,∞)\csc : \ \mathbb{R} \setminus \{k\pi\} \to (-\infty, -1] \cup [1, \infty)csc: R∖{kπ}→(−∞,−1]∪[1,∞)

Domain of cotangentcot⁡: R∖{kπ∣k∈Z}→R\cot : \ \mathbb{R} \setminus \{k\pi \mid k \in \mathbb{Z}\} \to \mathbb{R}cot: R∖{kπ∣k∈Z}→R

Domain of secantsec⁡: R∖{π2+kπ}→(−∞,−1]∪[1,∞)\sec : \ \mathbb{R} \setminus \left\{\tfrac{\pi}{2} + k\pi\right\} \to (-\infty, -1] \cup [1, \infty)sec: R∖{2π​+kπ}→(−∞,−1]∪[1,∞)

Domain of sine and cosinesin⁡,cos⁡: R→[−1,1]\sin, \cos : \ \mathbb{R} \to [-1, 1]sin,cos: R→[−1,1]

Domain of tangenttan⁡: R∖{π2+kπ | k∈Z}→R\tan : \ \mathbb{R} \setminus \left\{\tfrac{\pi}{2} + k\pi \,\middle|\, k \in \mathbb{Z}\right\} \to \mathbb{R}tan: R∖{2π​+kπ​k∈Z}→R

Sine: special valuessin⁡0=0,sin⁡π2=1,sin⁡π=0,sin⁡3π2=−1\sin 0 = 0, \quad \sin\tfrac{\pi}{2} = 1, \quad \sin\pi = 0, \quad \sin\tfrac{3\pi}{2} = -1sin0=0,sin2π​=1,sinπ=0,sin23π​=−1

Unit 3Differential Calculus

Derivative of 1 / x^n(shows up on exams)ddx 1xn=−nx n+1\frac{d}{dx}\, \frac{1}{x^{n}} = \frac{-n}{x^{\,n+1}}dxd​xn1​=xn+1−n​

Derivative of a constant(shows up on exams)ddx c=0\frac{d}{dx}\, c = 0dxd​c=0

Derivative of a^x(shows up on exams)ddx ax=(ln⁡a) ax\frac{d}{dx}\, a^{x} = (\ln a)\,a^{x}dxd​ax=(lna)ax

Derivative of arccos x(shows up on exams)ddx arccos⁡x=−11−x2\frac{d}{dx}\, \arccos x = -\frac{1}{\sqrt{1 - x^{2}}}dxd​arccosx=−1−x2​1​

Derivative of arccot x(shows up on exams)ddx arccot⁡x=−11+x2\frac{d}{dx}\, \operatorname{arccot} x = -\frac{1}{1 + x^{2}}dxd​arccotx=−1+x21​

Derivative of arcosh x(shows up on exams)ddx arcosh⁡x=1x2−1,∣x∣>1\frac{d}{dx}\, \operatorname{arcosh} x = \frac{1}{\sqrt{x^{2} - 1}}, \quad |x| > 1dxd​arcoshx=x2−1​1​,∣x∣>1

Derivative of arcoth x(shows up on exams)ddx arcoth⁡x=−1x2−1,∣x∣>1\frac{d}{dx}\, \operatorname{arcoth} x = -\frac{1}{x^{2} - 1}, \quad |x| > 1dxd​arcothx=−x2−11​,∣x∣>1

Derivative of arcsin x(shows up on exams)ddx arcsin⁡x=11−x2\frac{d}{dx}\, \arcsin x = \frac{1}{\sqrt{1 - x^{2}}}dxd​arcsinx=1−x2​1​

Derivative of arctan x(shows up on exams)ddx arctan⁡x=11+x2\frac{d}{dx}\, \arctan x = \frac{1}{1 + x^{2}}dxd​arctanx=1+x21​

Derivative of arsinh x(shows up on exams)ddx arsinh⁡x=11+x2\frac{d}{dx}\, \operatorname{arsinh} x = \frac{1}{\sqrt{1 + x^{2}}}dxd​arsinhx=1+x2​1​

Derivative of artanh x(shows up on exams)ddx artanh⁡x=11−x2,∣x∣<1\frac{d}{dx}\, \operatorname{artanh} x = \frac{1}{1 - x^{2}}, \quad |x| < 1dxd​artanhx=1−x21​,∣x∣<1

Derivative of cos x(shows up on exams)ddx cos⁡x=−sin⁡x\frac{d}{dx}\, \cos x = -\sin xdxd​cosx=−sinx

Derivative of cosh x(shows up on exams)ddx cosh⁡x=sinh⁡x\frac{d}{dx}\, \cosh x = \sinh xdxd​coshx=sinhx

Derivative of cot x(shows up on exams)ddx cot⁡x=−1sin⁡2x\frac{d}{dx}\, \cot x = -\frac{1}{\sin^{2} x}dxd​cotx=−sin2x1​

Derivative of coth x(shows up on exams)ddx coth⁡x=−1sinh⁡2x=1−coth⁡2x\frac{d}{dx}\, \coth x = -\frac{1}{\sinh^{2} x} = 1 - \coth^{2} xdxd​cothx=−sinh2x1​=1−coth2x

Derivative of e^x(shows up on exams)ddx ex=ex\frac{d}{dx}\, e^{x} = e^{x}dxd​ex=ex

Derivative of ln x(shows up on exams)ddx ln⁡x=1x\frac{d}{dx}\, \ln x = \frac{1}{x}dxd​lnx=x1​

Derivative of log_a x(shows up on exams)ddx log⁡ax=1x⋅ln⁡a\frac{d}{dx}\, \log_a x = \frac{1}{x \cdot \ln a}dxd​loga​x=x⋅lna1​

Derivative of sin x(shows up on exams)ddx sin⁡x=cos⁡x\frac{d}{dx}\, \sin x = \cos xdxd​sinx=cosx

Derivative of sinh x(shows up on exams)ddx sinh⁡x=cosh⁡x\frac{d}{dx}\, \sinh x = \cosh xdxd​sinhx=coshx

Derivative of tan x(shows up on exams)ddx tan⁡x=1cos⁡2x=1+tan⁡2x\frac{d}{dx}\, \tan x = \frac{1}{\cos^{2} x} = 1 + \tan^{2} xdxd​tanx=cos2x1​=1+tan2x

Derivative of tanh x(shows up on exams)ddx tanh⁡x=1cosh⁡2x=1−tanh⁡2x\frac{d}{dx}\, \tanh x = \frac{1}{\cosh^{2} x} = 1 - \tanh^{2} xdxd​tanhx=cosh2x1​=1−tanh2x

Derivative of the square root(shows up on exams)ddx x=12x\frac{d}{dx}\, \sqrt{x} = \frac{1}{2\sqrt{x}}dxd​x​=2x​1​

Power rule(shows up on exams)ddx xn=n⋅x n−1\frac{d}{dx}\, x^{n} = n \cdot x^{\,n-1}dxd​xn=n⋅xn−1

Unit 4Integral Calculus

Integral of 1 / x(shows up on exams)∫1x dx=ln⁡∣x∣+C\int \frac{1}{x} \, dx = \ln|x| + C∫x1​dx=ln∣x∣+C

Integral of a constant(shows up on exams)∫a dx=ax+C,a,C∈R\int a \, dx = a x + C, \quad a, C \in \mathbb{R}∫adx=ax+C,a,C∈R

Integral of a^x(shows up on exams)∫ax dx=1ln⁡a ax+C,a>0, a≠1\int a^{x} \, dx = \frac{1}{\ln a}\,a^{x} + C, \quad a > 0,\ a \neq 1∫axdx=lna1​ax+C,a>0, a=1

Integral of cot x(shows up on exams)∫cot⁡x dx=ln⁡∣sin⁡x∣+C\int \cot x \, dx = \ln|\sin x| + C∫cotxdx=ln∣sinx∣+C

Integral of e^{ax}(shows up on exams)∫eax dx=1a eax+C,a≠0\int e^{ax} \, dx = \frac{1}{a}\,e^{ax} + C, \quad a \neq 0∫eaxdx=a1​eax+C,a=0

Integral of e^x(shows up on exams)∫ex dx=ex+C\int e^{x} \, dx = e^{x} + C∫exdx=ex+C

Integral of tan x(shows up on exams)∫tan⁡x dx=−ln⁡∣cos⁡x∣+C\int \tan x \, dx = -\ln|\cos x| + C∫tanxdx=−ln∣cosx∣+C

Power rule for integration(shows up on exams)∫xa dx=1a+1 x a+1+C,a≠−1\int x^{a} \, dx = \frac{1}{a + 1}\,x^{\,a+1} + C, \quad a \neq -1∫xadx=a+11​xa+1+C,a=−1