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.

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

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

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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

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

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

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

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

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

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

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

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

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

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Rafia Shabbir

Calculus Formulas

Limit Formulas

L' Hospital Rule

Continuity Formulas

Continuous Function at a Point

Directional Continuity

Left-Continuous Function

Right-Continuous Function

Discontinuity

Removable Discontinuity

Jump Discontinuity

Essential Discontinuity

Derivative Formulas

Chain Rule

Derivative of an Implicit Function

Integrals Formulas

Integration by Parts

Integration by Substitution

Change of Variables

Definite Integral

Theory

Geometric Interpretation of the Derivative

Exponential Derivative

nth Derivative

Derivative

Basic Derivatives

Rate of Change

Derivatives

Derivatives of Trigonometric Functions

Inverse Function Rule

Physical Interpretation of the Derivative

Functional Power Rule

One-Sided Derivative

Derivative of an Implicit Function

Derivative Rules

Logarithmic Derivative

Derivatives of Inverse Trigonometric Functions

Chain Rule

Derivatives and Physics Word Problems

Differentials

Equation of the tangent line

Differentiability and Continuity

Product and Quotient Rules

Derivative Function

Formulas

Derivative Formulas

Exercises

Tangent Problems

Calculus Problems and Worksheets

Derivative Problems

Derivatives Worksheet

Differential Problems

Derivatives Worksheet

Derivatives Worksheet II

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