Formula Cheatsheet

Common formulas, rendered — with the LaTeX that makes them.

Algebra

Quadratic formula
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2-4ac}}{2a}
x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}
Difference of squares
a2−b2=(a+b)(a−b)a^2 - b^2 = (a+b)(a-b)
a^2 - b^2 = (a+b)(a-b)
Perfect square (sum)
(a+b)2=a2+2ab+b2(a+b)^2 = a^2 + 2ab + b^2
(a+b)^2 = a^2 + 2ab + b^2
Perfect square (difference)
(a−b)2=a2−2ab+b2(a-b)^2 = a^2 - 2ab + b^2
(a-b)^2 = a^2 - 2ab + b^2
Sum of cubes
a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a+b)(a^2 - ab + b^2)
a^3 + b^3 = (a+b)(a^2 - ab + b^2)
Difference of cubes
a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a-b)(a^2 + ab + b^2)
a^3 - b^3 = (a-b)(a^2 + ab + b^2)
Exponent product rule
am⋅an=am+na^m \cdot a^n = a^{m+n}
a^m \cdot a^n = a^{m+n}
Exponent power rule
(am)n=amn(a^m)^n = a^{mn}
(a^m)^n = a^{mn}
Logarithm of a product
log⁡b(xy)=log⁡bx+log⁡by\log_b(xy) = \log_b x + \log_b y
\log_b(xy) = \log_b x + \log_b y
Change of base
log⁡bx=log⁡kxlog⁡kb\log_b x = \frac{\log_k x}{\log_k b}
\log_b x = \frac{\log_k x}{\log_k b}

Geometry

Circle area
A=πr2A = \pi r^2
A = \pi r^2
Circle circumference
C=2πrC = 2\pi r
C = 2\pi r
Triangle area
A=12bhA = \tfrac{1}{2} b h
A = \tfrac{1}{2} b h
Rectangle area
A=lwA = l w
A = l w
Trapezoid area
A=12(b1+b2)hA = \tfrac{1}{2}(b_1+b_2)h
A = \tfrac{1}{2}(b_1+b_2)h
Sphere volume
V=43πr3V = \tfrac{4}{3}\pi r^3
V = \tfrac{4}{3}\pi r^3
Sphere surface area
A=4πr2A = 4\pi r^2
A = 4\pi r^2
Cylinder volume
V=πr2hV = \pi r^2 h
V = \pi r^2 h
Cone volume
V=13πr2hV = \tfrac{1}{3}\pi r^2 h
V = \tfrac{1}{3}\pi r^2 h
Pythagorean theorem
a2+b2=c2a^2 + b^2 = c^2
a^2 + b^2 = c^2
Distance between two points
d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}
d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}
Slope of a line
m=y2−y1x2−x1m = \frac{y_2-y_1}{x_2-x_1}
m = \frac{y_2-y_1}{x_2-x_1}

Trigonometry

Pythagorean identity
sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1
\sin^2\theta + \cos^2\theta = 1
Tangent
tan⁡θ=sin⁡θcos⁡θ\tan\theta = \frac{\sin\theta}{\cos\theta}
\tan\theta = \frac{\sin\theta}{\cos\theta}
Law of sines
asin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
Law of cosines
c2=a2+b2−2abcos⁡Cc^2 = a^2 + b^2 - 2ab\cos C
c^2 = a^2 + b^2 - 2ab\cos C
Sine of a sum
sin⁡(a+b)=sin⁡acos⁡b+cos⁡asin⁡b\sin(a+b) = \sin a \cos b + \cos a \sin b
\sin(a+b) = \sin a \cos b + \cos a \sin b
Cosine of a sum
cos⁡(a+b)=cos⁡acos⁡b−sin⁡asin⁡b\cos(a+b) = \cos a \cos b - \sin a \sin b
\cos(a+b) = \cos a \cos b - \sin a \sin b
Double angle (sine)
sin⁡(2θ)=2sin⁡θcos⁡θ\sin(2\theta) = 2\sin\theta\cos\theta
\sin(2\theta) = 2\sin\theta\cos\theta
Double angle (cosine)
cos⁡(2θ)=cos⁡2θ−sin⁡2θ\cos(2\theta) = \cos^2\theta - \sin^2\theta
\cos(2\theta) = \cos^2\theta - \sin^2\theta

Calculus

Power rule (derivative)
ddxxn=nxn−1\frac{d}{dx}x^n = nx^{n-1}
\frac{d}{dx}x^n = nx^{n-1}
Product rule
ddx[fg]=f′g+fg′\frac{d}{dx}[fg] = f'g + fg'
\frac{d}{dx}[fg] = f'g + fg'
Quotient rule
ddx ⁣[fg]=f′g−fg′g2\frac{d}{dx}\!\left[\frac{f}{g}\right] = \frac{f'g - fg'}{g^2}
\frac{d}{dx}\!\left[\frac{f}{g}\right] = \frac{f'g - fg'}{g^2}
Chain rule
ddxf(g(x))=f′(g(x)) g′(x)\frac{d}{dx}f(g(x)) = f'(g(x))\,g'(x)
\frac{d}{dx}f(g(x)) = f'(g(x))\,g'(x)
Derivative of e^x
ddxex=ex\frac{d}{dx}e^x = e^x
\frac{d}{dx}e^x = e^x
Derivative of ln x
ddxln⁡x=1x\frac{d}{dx}\ln x = \frac{1}{x}
\frac{d}{dx}\ln x = \frac{1}{x}
Derivative of a constant
ddxc=0\frac{d}{dx}c = 0
\frac{d}{dx}c = 0
Derivative of x
ddxx=1\frac{d}{dx}x = 1
\frac{d}{dx}x = 1
Derivative of sin x
ddxsin⁡x=cos⁡x\frac{d}{dx}\sin x = \cos x
\frac{d}{dx}\sin x = \cos x
Derivative of cos x
ddxcos⁡x=−sin⁡x\frac{d}{dx}\cos x = -\sin x
\frac{d}{dx}\cos x = -\sin x
Derivative of tan x
ddxtan⁡x=sec⁡2x\frac{d}{dx}\tan x = \sec^2 x
\frac{d}{dx}\tan x = \sec^2 x
Example: derivative of x^3
ddxx3=3x2\frac{d}{dx}x^3 = 3x^2
\frac{d}{dx}x^3 = 3x^2
Example: derivative of a polynomial
ddx(5x2+3x−7)=10x+3\frac{d}{dx}(5x^2 + 3x - 7) = 10x + 3
\frac{d}{dx}(5x^2 + 3x - 7) = 10x + 3
Power rule (integral)
∫xn dx=xn+1n+1+C\int x^n \, dx = \frac{x^{n+1}}{n+1} + C
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C
Integral of a constant
∫k dx=kx+C\int k \, dx = kx + C
\int k \, dx = kx + C
Integral of sin x
∫sin⁡x dx=−cos⁡x+C\int \sin x \, dx = -\cos x + C
\int \sin x \, dx = -\cos x + C
Integral of cos x
∫cos⁡x dx=sin⁡x+C\int \cos x \, dx = \sin x + C
\int \cos x \, dx = \sin x + C
Integral of 1/x
∫1x dx=ln⁡∣x∣+C\int \frac{1}{x} \, dx = \ln|x| + C
\int \frac{1}{x} \, dx = \ln|x| + C
Integral of e^x
∫ex dx=ex+C\int e^x \, dx = e^x + C
\int e^x \, dx = e^x + C
Example: integral of 3x^2
∫3x2 dx=x3+C\int 3x^2 \, dx = x^3 + C
\int 3x^2 \, dx = x^3 + C
Example: integral of a linear expression
∫(2x+1) dx=x2+x+C\int (2x + 1) \, dx = x^2 + x + C
\int (2x + 1) \, dx = x^2 + x + C
Fundamental theorem of calculus
∫abf′(x) dx=f(b)−f(a)\int_a^b f'(x)\,dx = f(b) - f(a)
\int_a^b f'(x)\,dx = f(b) - f(a)

Statistics & Probability

Mean
xˉ=1n∑i=1nxi\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i
\bar{x} = \frac{1}{n}\sum_{i=1}^{n} x_i
Variance
σ2=1n∑i=1n(xi−xˉ)2\sigma^2 = \frac{1}{n}\sum_{i=1}^{n} (x_i - \bar{x})^2
\sigma^2 = \frac{1}{n}\sum_{i=1}^{n} (x_i - \bar{x})^2
Standard deviation
σ=σ2\sigma = \sqrt{\sigma^2}
\sigma = \sqrt{\sigma^2}
Permutations
P(n,r)=n!(n−r)!P(n,r) = \frac{n!}{(n-r)!}
P(n,r) = \frac{n!}{(n-r)!}
Combinations
C(n,r)=(nr)=n!r!(n−r)!C(n,r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}
C(n,r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}
Conditional probability
P(A∣B)=P(A∩B)P(B)P(A \mid B) = \frac{P(A \cap B)}{P(B)}
P(A \mid B) = \frac{P(A \cap B)}{P(B)}
Bayes' theorem
P(A∣B)=P(B∣A) P(A)P(B)P(A \mid B) = \frac{P(B \mid A)\,P(A)}{P(B)}
P(A \mid B) = \frac{P(B \mid A)\,P(A)}{P(B)}

Arithmetic & Numbers

Percentage change
%Δ=new−oldold×100\%\Delta = \frac{\text{new} - \text{old}}{\text{old}} \times 100
\%\Delta = \frac{\text{new} - \text{old}}{\text{old}} \times 100
Simple interest
I=PrtI = Prt
I = Prt
Compound interest
A=P(1+rn)ntA = P\left(1+\frac{r}{n}\right)^{nt}
A = P\left(1+\frac{r}{n}\right)^{nt}
Arithmetic sequence (nth term)
an=a1+(n−1)da_n = a_1 + (n-1)d
a_n = a_1 + (n-1)d
Geometric sequence (nth term)
an=a1⋅rn−1a_n = a_1 \cdot r^{n-1}
a_n = a_1 \cdot r^{n-1}
Formula Cheatsheet — MathSnap