Helpful Formulas for Calculus (Know Them!) - Vocabulary Flashcards

A set of vocabulary-style flashcards covering limits, derivatives, and integrals from the provided lecture notes.

d/dx [x^n]

n x^{n-1} (n \neq -1).

d/dx [c f(x)]

c f'(x).

d/dx [f(x) g(x)]

f'(x) g(x) + f(x) g'(x).

d/dx \left[\frac{f(x)}{g(x)}\right]

\frac{f'(x) g(x) - f(x) g'(x)}{[g(x)]^2}.

d/dx [f(g(x))]

f'(g(x)) \cdot g'(x).

d/dx [\sin x]

\cos x.

d/dx [\cos x]

-\sin x.

d/dx [\tan x]

\sec^2 x.

d/dx [\cot x]

-\csc^2 x.

d/dx [\sec x]

\sec x \tan x.

d/dx [\csc x]

-\csc x \cot x.

d/dx [\ln x]

\frac{1}{x}.

d/dx [\log_b x]

\frac{1}{x \ln b}.

d/dx [\arcsin x]

\frac{1}{\sqrt{1 - x^2}}.

d/dx [\arccos x]

-\frac{1}{\sqrt{1 - x^2}}.

d/dx [\arctan x]

\frac{1}{1 + x^2}.

d/dx [\arcsec x]

\frac{1}{|x| \sqrt{x^2 - 1}}.

d/dx [\arccsc x]

-\frac{1}{|x| \sqrt{x^2 - 1}}.

ILATE

\text{Inverse Trig, Logarithmic, Algebraic, Trigonometric, Exponential (order for choosing } u\text{)}.

\int dx

x + C.

\int \sec x dx

\ln|\sec x + \tan x| + C.

\int \csc x dx

-\ln|\csc x + \cot x| + C.

\int \frac{1}{1 + x^2} dx

\arctan x + C.

\int \frac{1}{x} dx

\ln|x| + C.

\int \sec x \tan x dx

\sec x + C.

\int \csc x \cot x dx

-\csc x + C.

\int \frac{1}{a x + b} dx

\frac{1}{a} \ln|a x + b| + C.

\int \sqrt{x} dx

\frac{2}{3} x^{3/2} + C.

\int x^n dx

\frac{x^{n+1}}{n+1} + C \quad (n \neq -1).

\int e^x dx

e^x + C.

\sin(0)

( 0^{\circ}): y=0

\cos(0)

( 0^{\circ}): x=1

\tan(0)

( 0^{\circ}): 0

\sin(\pi/6)

( 30^{\circ}): \frac{1}{2}

\cos(\pi/6)

( 30^{\circ}): \frac{\sqrt{3}}{2}

\tan(\pi/6)

( 30^{\circ} ): \frac{\sqrt{3}}{3}

\sin(\pi/4)

( 45^{\circ} ): \frac{\sqrt{2}}{2}

\cos(\pi/4)

( 45^{\circ}): \frac{\sqrt{2}}{2}

\tan(\pi/4)

( 45^{\circ}): 1

\sin(\pi/3)

( 60^{\circ}): \frac{\sqrt{3}}{2} =y

\cos(\pi/3)

( 60^{\circ} ): \frac{1}{2} =x

\tan(\pi/3)

( 60^{\circ}): \sqrt{3}

\sin(\pi/2)

( 90^{\circ}): 1

\cos(\pi/2)

( 90^{\circ}): 0

\tan(\pi/2)

( 90^{\circ} ): Undefined