← Calculus

formula sheet

every formula from the calculus notes in one place

Pre-calc reference

the algebra, log rules, and trig identities these formulas assume

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Limits

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Limit \(\lim\limits_{x \to a} f(x) = L\)
Existence \(\lim\limits_{x \to a^-} f(x) = \lim\limits_{x \to a^+} f(x) = L\)
Continuity at \(a\) \(\lim\limits_{x \to a} f(x) = f(a)\)
Sum and difference \(\lim (f \pm g) = \lim f \pm \lim g\)
Constant multiple \(\lim c\,f = c \lim f\)
Product \(\lim fg = \lim f \cdot \lim g\)
Quotient \(\lim \dfrac{f}{g} = \dfrac{\lim f}{\lim g}\), \(\lim g \neq 0\)
Power and root \(\lim [f]^n = [\lim f]^n\)
Composition \(\lim f(g(x)) = f(\lim g(x))\), \(f\) continuous

Limits at infinity

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Top degree \(<\) bottom \(\lim\limits_{x \to \infty} \dfrac{p(x)}{q(x)} = 0\)
Degrees equal ratio of the leading coefficients
Top degree \(>\) bottom \(\pm\infty\)
Growth ranking \(\ln x \ll x^n \ll e^x \ll x!\)
Standard move divide every term by the highest power of \(x\)

Indeterminate forms & l'Hôpital

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The seven forms

Quotient \(\dfrac{0}{0}\), \(\dfrac{\infty}{\infty}\)
Product and difference \(0 \cdot \infty\), \(\infty - \infty\)
Exponential \(1^\infty\), \(0^0\), \(\infty^0\)
Not indeterminate \(\dfrac{k}{0} = \pm\infty\), \(\dfrac{k}{\infty} = 0\), \(0^\infty = 0\)

L'Hôpital's rule

The rule \(\lim \dfrac{f(x)}{g(x)} = \lim \dfrac{f'(x)}{g'(x)}\)
Condition only for \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), rechecked each pass
\(0 \cdot \infty\) rewrite \(fg\) as \(\dfrac{f}{1/g}\)
Exponential forms \(y = f^g \Rightarrow \lim y = e^{\lim g \ln f}\)
Classic result \(\lim\limits_{x \to \infty}\left(1 + \tfrac{1}{x}\right)^x = e\)

Differentiation rules

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Definition \(f'(x) = \lim\limits_{h \to 0} \dfrac{f(x+h) - f(x)}{h}\)
Power rule \(\dfrac{d}{dx}x^n = nx^{n-1}\)
Constant \(\dfrac{d}{dx}c = 0\)
Constant multiple \(\dfrac{d}{dx}c\,f = c\,f'\)
Sum and difference \((f \pm g)' = f' \pm g'\)
Product rule \((uv)' = u'v + uv'\)
Quotient rule \(\left(\dfrac{u}{v}\right)' = \dfrac{u'v - uv'}{v^2}\)
Chain rule \(\dfrac{d}{dx}f(g(x)) = f'(g(x))\,g'(x)\)
Chain rule, Leibniz \(\dfrac{dy}{dx} = \dfrac{dy}{du}\cdot\dfrac{du}{dx}\)
Implicit \(\dfrac{d}{dx}y^n = ny^{n-1}\dfrac{dy}{dx}\)
Tangent line at \(a\) \(y - f(a) = f'(a)(x - a)\)

Derivatives of common functions

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Trigonometric

\(\sin x\) \(\cos x\)
\(\cos x\) \(-\sin x\)
\(\tan x\) \(\sec^2 x\)
\(\sec x\) \(\sec x \tan x\)
\(\csc x\) \(-\csc x \cot x\)
\(\cot x\) \(-\csc^2 x\)

Exponential and logarithmic

\(e^x\) \(e^x\)
\(b^x\) \(b^x \ln b\)
\(\ln x\) \(\dfrac{1}{x}\)
\(\ln|x|\) \(\dfrac{1}{x}\)
\(\log_b x\) \(\dfrac{1}{x \ln b}\)

Inverse trigonometric

\(\arcsin x\) \(\dfrac{1}{\sqrt{1 - x^2}}\)
\(\arccos x\) \(-\dfrac{1}{\sqrt{1 - x^2}}\)
\(\arctan x\) \(\dfrac{1}{1 + x^2}\)
\(\operatorname{arccot} x\) \(-\dfrac{1}{1 + x^2}\)

With the chain rule

\([g(x)]^n\) \(n[g(x)]^{n-1}g'(x)\)
\(e^{g(x)}\) \(e^{g(x)}g'(x)\)
\(\ln g(x)\) \(\dfrac{g'(x)}{g(x)}\)
\(\sin g(x)\) \(\cos(g(x))\,g'(x)\)

Integration rules

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Antiderivative \(\displaystyle\int f(x)\,dx = F(x) + C\), \(F' = f\)
Power rule \(\displaystyle\int x^n dx = \dfrac{x^{n+1}}{n+1} + C\), \(n \neq -1\)
Constant multiple \(\displaystyle\int c\,f\,dx = c\int f\,dx\)
Sum and difference \(\displaystyle\int (f \pm g)\,dx = \int f\,dx \pm \int g\,dx\)
Fundamental theorem \(\displaystyle\int_a^b f(x)\,dx = F(b) - F(a)\)
Accumulation \(\dfrac{d}{dx}\displaystyle\int_a^x f(t)\,dt = f(x)\)
Net change \(\displaystyle\int_a^b f'(x)\,dx = f(b) - f(a)\)
Swapping limits \(\displaystyle\int_b^a f = -\int_a^b f\), and \(\displaystyle\int_a^a f = 0\)
Splitting \(\displaystyle\int_a^c f = \int_a^b f + \int_b^c f\)
u-substitution \(\displaystyle\int f(g(x))g'(x)\,dx = \int f(u)\,du\)
Integration by parts \(\displaystyle\int u\,dv = uv - \int v\,du\)
By parts, definite \(\displaystyle\int_a^b u\,dv = \Big[uv\Big]_a^b - \int_a^b v\,du\)
Linear inside \(\displaystyle\int f(ax+b)\,dx = \dfrac{1}{a}F(ax+b) + C\)
LIATE log, inverse trig, algebraic, trig, exponential; first one is \(u\)

Common integrals

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Powers, exponentials, logarithms

\(\displaystyle\int k\,dx\) \(kx + C\)
\(\displaystyle\int x^n dx\) \(\dfrac{x^{n+1}}{n+1} + C\), \(n \neq -1\)
\(\displaystyle\int \frac{1}{x}dx\) \(\ln|x| + C\)
\(\displaystyle\int e^x dx\) \(e^x + C\)
\(\displaystyle\int b^x dx\) \(\dfrac{b^x}{\ln b} + C\)
\(\displaystyle\int \ln x\,dx\) \(x\ln x - x + C\)

Trigonometric

\(\displaystyle\int \sin x\,dx\) \(-\cos x + C\)
\(\displaystyle\int \cos x\,dx\) \(\sin x + C\)
\(\displaystyle\int \sec^2 x\,dx\) \(\tan x + C\)
\(\displaystyle\int \sec x \tan x\,dx\) \(\sec x + C\)
\(\displaystyle\int \tan x\,dx\) \(-\ln|\cos x| + C\)
\(\displaystyle\int \sec x\,dx\) \(\ln|\sec x + \tan x| + C\)

Forms giving inverse trig

\(\displaystyle\int \frac{dx}{1 + x^2}\) \(\arctan x + C\)
\(\displaystyle\int \frac{dx}{\sqrt{1 - x^2}}\) \(\arcsin x + C\)
\(\displaystyle\int \frac{dx}{a^2 + x^2}\) \(\dfrac{1}{a}\arctan\dfrac{x}{a} + C\)
\(\displaystyle\int \frac{dx}{\sqrt{a^2 - x^2}}\) \(\arcsin\dfrac{x}{a} + C\)