Chapters

Limit Formulas
L' Hospital Rule
Continuity Formulas
Left-Continuous Function
Right-Continuous Function
Removable Discontinuity
Jump Discontinuity
Essential Discontinuity
Derivative Formulas
Chain Rule
Derivative of an Implicit Function
Integrals Formulas

Limit Formulas

$\lim _ {x \rightarrow a} k = k$

$\lim _ {x \rightarrow a} [ f (x) \pm g (x) ] = \lim _ { x \rightarrow a} f(x) \pm \lim _ { x \rightarrow a} g(x)$

$\lim _ {x \rightarrow a} [ f (x) \cdot g (x) ] = \lim _ { x \rightarrow a} f(x) \cdot \lim _ { x \rightarrow a} g(x)$

$\lim _ {x \rightarrow a} \frac { f(x) } { g (x)} = \frac { \lim_ {x \rightarrow a} f(x) } { \lim _{x \rightarrow a} g(x) } if \lim _ {x \rightarrow a} g(x) \neq 0$

$\lim_ {x \rightarrow a} [f (x) ^ {g (x)}] = \lim_ {x \rightarrow} [f(x) ] ^ {\lim _ {x \rightarrow a} g(x)} if f(x) >0$

$\lim_ {x \rightarrow a} g[f (x)] = g [ \lim _ {x \rightarrow a} f(x) ]$

g can be a root, a log, sin, cos, tan, etc.

$\lim_ {x \rightarrow a} \sqrt [n] {f (x)} = \sqrt [n] { \lim_ {x \rightarrow a} f(x)}$

$\lim _ {x \rightarrow a} [log _ a f(x)] = log _a [\lim_ {x \rightarrow a} f(x) ] if a >0 and f(x) > 0$

L' Hospital Rule

$\lim _ { x \rightarrow a} \frac {f(x)} {g (x)} = \lim _ { x \rightarrow a} \frac {f ' (x)} {g ' (x)}$

Continuity Formulas

Continuous Function at a Point

$\ni f (a)$

$\ni \lim _ {x \rightarrow a} f(x) \leftrightarrow \lim _ {x \rightarrow a ^ {-}} f(x) = \lim_ { x \rightarrow a^ {+}} f(x)$

$f(a) = \lim {x \rightarrow a} f(x)$

Directional Continuity

Left-Continuous Function

$f(a) = \lim _ {x \rightarrow a ^ {-}} f(x)$

Right-Continuous Function

$f(a) = \lim _ {x \rightarrow a ^ {+}} f(x)$

Discontinuity

Removable Discontinuity

$∃ f (a) or f(a) \neq \lim _ {x \rightarrow a} f(x)$

Jump Discontinuity

$\lim _ {x \rightarrow a ^ {-}} f(x) \neq \lim _ {x \rightarrow a ^ {+}} f(x)$

Essential Discontinuity

Derivative Formulas

u and v are functions of x.

a, e and k are constants (real numbers)

$f(x) = k$ $f ' (x) = 0$

$f(x) = ku$ $f ' (x) = k u'$

$f(x) = x$ $f ' (x) = 1$

$f(x) = u ^ k$ $f ' (x) = k \cdot u ^ {k - 1} \cdot u'$

$f(x) = \sqrt{u}$ $f ' (x) = \frac { u'} {2 \cdot \sqrt{u}}$

$f(x) = \sqrt [k] {u}$ $f ' (x) = \frac { u '} {k \cdot \sqrt [k] {u ^ {k -1}}}$

$f(x) = u \pm v$ $f ' (x) = u ' \pm v'$

$f(x) = u \cdot v$ $f ' (x) = u ' \cdot v + u \cdot v'$

$f(x) = \frac {u} {v}$ $f ' (x) = \frac {u' v - u v'} {v ^ 2}$

$f(x) = \frac {k } {v}$ $f ' (x) = \frac {-k \cdot v ^2} { v ^2}$

$f(x) = \frac {1} {v}$ $f ' (x) = \frac {- v '} {v ^2}$

$f(x) = a ^ u$ $f ' (x) = u' \cdot a ^ u\cdot lna$

$f(x) = e ^ u$ $f' (x) = u' \cdot e ^ u$

$f (x) log_a u$ $f ' (x) = \frac {u '} {u \cdot lna} = \frac {u '} {u} \cdot log _a e = \frac {u'}{u} \cdot \frac {1} {lna}$

$f(x) = lnu$ $f ' (x) = \frac {u'} {u}$

$f (x) = sin u$ $f ' (x) = u ' \cdot cos u$

$f(x) = cos u$ $f ' (x) = - u' \cdot sin u$

$f(x) = tan u$ $f ' (x) = \frac {u '} {cos^2 u} = u ' \cdot sec ^2 u = u ' \cdot (1 + tan ^2 u)$

$f(x) = cot u$ $f ' (x) = - \frac {u '} {sen ^2 u } = - u' \cdot c sc ^2 u = - u' \cdot (1 + cot ^2 u)$

$f(x) = sec u$ $f ' (x) = \frac {u ' \cdot sin u} { cos ^2 u} = u ' \cdot sec u \cdot tan u$

$f(x) = csc u$ $f ' (x) = - \frac {u ' \cdot cos u} {sin ^2 u} = - u' \cdot csc u \cdot cot u$

$f(x) = arcsin u$ $f ' (x) = \frac { u'} { \sqrt {1 - u ^2}}$

$f(x) = arccos u$ $f ' (x) = - \frac {u '} { \sqrt {1 - u ^2}}$

$f(x) = arctan u$ $f ' (x) = \frac { u '} { 1 = u ^2}$

$f (x) = arc cot u$ $f ' (x) = - \frac {u '} { 1 + u ^2}$

$f(x) = arc sec u$ $f ' (x) = \frac { u '} {u \cdot \sqrt {u ^2 - 1}}$

$f(x) = arocsc u$ $f ' (x) = - \frac {u '} {u \cdot \sqrt {u ^2 - 1}}$

$f(x) = u ^v$ $f ' (x) = v \cdot u ^ {v - 1} \cdot u ' + u ^v \cdot v' \cdot ln u$

Chain Rule

$(g o f) ' (x) = g ' [f(x)] \cdot f ' (x)$

Derivative of an Implicit Function

$y' = \frac { F ' _x} {F ' _y}$

Integrals Formulas

$\int dx = x + C$

$\int k dx = k \cdot x + C$

$\int u ^n \cdot u ' dx = \frac { u ^ {n + 1}} {n + 1} + C$ $n \neq -1$

$\int \frac {u '} {u} dx = ln u + C$

$\int a ^ u \cdot u ' dx = \frac { a ^ u} { ln a} + C$

$\int e ^ u \cdot u' dx = e ^ u + C$

$\int sin u\cdot u ' dx = -cos u + C$

$\int cos u \cdot u ' dx = sin u + C$

$\int \frac { u ' } { cos ^2 u } dx = \int sec ^2 u \cdot u ' dx = \int (1 + tan^2 u) \cdot u ' dx = tan u + C$

$\int \frac { u '} {sen ^2 u} dx = \int cosec ^ 2 u \cdot u ' dx = \int (1 + cot g ^2 u) \cdot u ' dx = -cotg u + C$

$\int \frac { u ' } { \sqrt { 1 - u ^2}} dx = arcsin u + C$

$\int \frac {u ' } { 1 + u ^2} dx = arc tan u + C$

Integration by Parts

$\int u \cdot v ' dx = u \cdot v - \int u ' \cdot v dx$

Integration by Substitution

$\int f ' (u) \cdot u' dx = F (u) + C$

Change of Variables

$\int R (x , \sqrt {a ^2 - x ^2}) dx$ $x =a sin t$

$\int R (x , \sqrt {a ^2 + x ^2}) dx$ $x =a tan t$

$\int R (x , \sqrt {x ^2 - a ^2}) dx$ $x =a sec t$

$\int R ( x , \sqrt [n] {\frac {ax + b} {cx + d}}) dx$ $t = \frac {ax + b} {cx - d}$

If $R( sinx , cos x)$ is even

Change $t = tan x$

sin x $\frac {t} {\sqrt {1 + t ^2}}$

cos x $\frac {1} {\sqrt {1 + t ^2}}$

tan x t

dx $\frac {dt} {\sqrt {1 + t ^2}}$

If $R( sinx , cos x)$ is not even:

Change $tg\frac {x}{2}$

sen x $\frac {2t} {1 + t ^2}$

cos x $\frac {1 - t ^2} {1 + t ^2}$

tgx $\frac {2t} {1 - t ^2}$

dx $\frac {2dt} {1 + t ^2}$

Definite Integral

$\int _{a} ^ {b} f(x) = [F (x) ]_{a} ^{b} = F (b) - F(a)$

$\int _ {a} ^ {b} f(x) dx = - \int _{b} ^ {a} f(x) dx$

$\int _ {a} ^ {a} f(x) dx = 0$

$\int _ {b} {a} f(x) dx = \int_ {c} ^ {a} f(x) dx + \int _ {c} ^ {b} f(x) dx$

$\int _{a} ^ {b} [ f (x) + g(x) ^ 1 dx = \int _ {a} {b} f(x) dx + \int _ {a} ^ {b} g(x) dx$

$\int _ {a} ^ {b} \cdot f(x) dx = k \cdot \int _ {a} ^ {b} f(x) dx$

$\int _ {a} ^ {b} f(x) dx = (b - a) \cdot f(c)$

$A = \int_ {a} ^ {b{ f(x) dx$

$V = \pi \int_ {a} ^ {b} [ f (x) ] ^2 dx$

Did you like the article?

(1 votes, average: 5.00 out of 5)

Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.

Did you like
this resource?

Bravo!

Download it in pdf format by simply entering your e-mail!

Calculus Formulas

Limit Formulas

L' Hospital Rule