Calculus Reference

Definitions

Critical point

f^{'} (c) = 0 or DNE

A domain point where $f^{'} (x) = 0$ or $f^{'} (x)$ does not exist.

Example:

x = 0

is critical for

f (x) = x^{2}

Inflection point

A point where the graph changes concavity.

Example: The origin is an inflection point of

y = x^{3}

Local extremum

A maximum or minimum compared with nearby function values.

Example:

y = (x - 2)^{2}

has a local minimum at

x = 2

Absolute extremum

The greatest or least function value on the full stated domain or interval.

Example: On

[- 1, 2]

x^{2}

has an absolute maximum of

4

Continuity

x \to a lim f (x) = f (a)

A function is continuous at $a$ when its limit exists, $f (a)$ exists, and the two are equal.

Example: Polynomials are continuous for every

x \in R

One-sided limit

x \to a^{-} lim f (x)

The value approached from only the left or only the right.

Example: A jump can have two different one-sided limits.

Limit at infinity

x \to \infty lim f (x)

The value or behaviour approached as $x$ grows without bound.

Example:

\frac{1}{x}

approaches

0

x \to \infty

Infinite limit

x \to a lim f (x) = \infty

Function values grow without bound as $x$ approaches a finite input.

Example:

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

grows to

\infty

x \to 0

Removable discontinuity

A missing point where a finite two-sided limit exists.

Example: A cancelled rational factor often creates a hole.

Jump discontinuity

A break where finite left- and right-hand limits exist but differ.

Example: A step function jumps at its boundary.

Infinite discontinuity

A break associated with unbounded behaviour and usually a vertical asymptote.

Example:

\frac{1}{x - 2}

has an infinite discontinuity at

x = 2

Formula sheet

15 formulas

Power rule

\frac{d}{d x} x^{n} = n x^{n - 1}

Variables: $n$ : real constant where the expression is defined
Remember: Multiply by the exponent, then reduce it by one.

Higher derivatives

f^{''} (x) = \frac{d ^{2} y}{d x ^{2}}

Variables: $f^{''}$ : second derivative
Remember: Measures how the first derivative changes.

Product rule

(f g)^{'} = f^{'} g + f g^{'}

Variables: $f, g$ : differentiable functions
Remember: Differentiate one factor at a time.

Quotient rule

(\frac{f}{g})^{'} = \frac{f ^{'} g - f g ^{'}}{g ^{2}}

Variables: $g \neq = 0$
Remember: Low d-high minus high d-low, over low squared.

Chain rule

\frac{d}{d x} f (g (x)) = f^{'} (g (x)) g^{'} (x)

Variables: outer $f$ ; inner $g$
Remember: Derivative of the outside times derivative of the inside.

Implicit differentiation

\frac{d}{d x} F (x, y) = 0

Variables: $y$ is treated as a function of $x$
Remember: Every derivative of a $y$ -term receives a factor $\frac{d y}{d x}$ .

Exponential and log derivatives

\frac{d}{d x} e^{x} = e^{x}, \frac{d}{d x} ln x = \frac{1}{x}

Variables: $x > 0$ for $ln x$
Remember: Use chain rule for composite exponents or log arguments.

Trig derivatives

(sin x)^{'} = cos x, (cos x)^{'} = - sin x

Variables: $x$ in radians
Remember: The standard rules assume radians.

Basic antiderivative

\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C

Variables: $n \neq = - 1$ ; $C$ : constant
Remember: Increase the exponent, then divide.

Log antiderivative

\int \frac{1}{x} d x = ln ∣ x ∣ + C

Variables: $x \neq = 0$
Remember: Absolute value covers positive and negative intervals.

Definite integral

\int_{a}^{b} f (x) d x = F (b) - F (a)

Variables: $F^{'} = f$
Remember: Signed accumulation between bounds.

Substitution

\int f (g (x)) g^{'} (x) d x = \int f (u) d u

Variables: $u = g (x)$
Remember: Reverse the chain rule.

Integration by parts

\int u d v = uv - \int v d u

Variables: $u, d v$ : chosen factors
Remember: Reverse the product rule.

Partial fractions

\frac{P ( x )}{( x - a ) ( x - b )} = \frac{A}{x - a} + \frac{B}{x - b}

Variables: distinct linear factors shown
Remember: Decompose a proper rational expression before integrating.

Exponential integrals

\int e^{k x} d x = \frac{1}{k} e^{k x} + C

Variables: $k \neq = 0$
Remember: Divide by the inner derivative.

Theorem conditions

Fermat’s Theorem

Assumptions: $f$ has a local extremum at interior point $c$ and is differentiable there.
Conclusion: $f^{'} (c) = 0$ .
In one line: Smooth interior extrema must have a horizontal tangent.

Extreme Value Theorem

Assumptions: $f$ is continuous on the closed interval $[a, b]$ .
Conclusion: $f$ attains an absolute maximum and minimum on $[a, b]$ .
In one line: A continuous closed-interval graph cannot miss its highest or lowest value.

First Derivative Test

Assumptions: $c$ is critical and $f^{'}$ changes sign around $c$ .
Conclusion: $+ \to -$ gives a local maximum; $- \to +$ gives a local minimum.
In one line: Track whether the function switches from rising to falling or vice versa.

Second Derivative Test

Assumptions: $f^{'} (c) = 0$ and $f^{''}$ exists near $c$ .
Conclusion: $f^{''} (c) > 0$ gives a local minimum; $f^{''} (c) < 0$ gives a local maximum.
In one line: Concavity classifies a horizontal critical point when $f^{''} (c) \neq = 0$ .

Mean Value Theorem

Assumptions: $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$ .
Conclusion: Some $c$ satisfies $f^{'} (c) = \frac{f ( b ) - f ( a )}{b - a}$ .
In one line: At some point, instantaneous change matches average change.

Rolle’s Theorem

Assumptions: Mean Value Theorem conditions hold and $f (a) = f (b)$ .
Conclusion: Some $c \in (a, b)$ has $f^{'} (c) = 0$ .
In one line: A smooth trip returning to the same height has a horizontal tangent somewhere.

Squeeze Theorem

Assumptions: $g (x) \leq f (x) \leq h (x)$ near $a$ , and $g, h$ approach the same limit $L$ .
Conclusion: $f (x)$ also approaches $L$ .
In one line: A function trapped between two converging functions must share their limit.

Fundamental Theorem of Calculus

Assumptions: $f$ is continuous and $F$ is an antiderivative of $f$ .
Conclusion: The definite integral equals $F (b) - F (a)$ , and accumulation differentiates back to $f$ .
In one line: Differentiation and integration are inverse processes.

L’Hôpital’s Rule

Assumptions: A differentiable quotient has indeterminate form $\frac{0}{0}$ or $\frac{\infty}{\infty}$ , and the rule conditions hold.
Conclusion: The quotient limit may equal the limit of $\frac{f ^{'}}{g ^{'}}$ .
In one line: Compare rates of change when direct substitution is indeterminate.

Worked examples

Differentiate a composite

Differentiate $y = (3 x^{2} + 1)^{4}$ .

Show solution

Outer function: $u^{4}$ has derivative $4 u^{3}$ .
Inner function: $u = 3 x^{2} + 1$ has derivative $6 x$ .
Chain rule: $y^{'} = 4 (3 x^{2} + 1)^{3} (6 x)$ .
Simplify: $y^{'} = 24 x (3 x^{2} + 1)^{3}$ .

Variable glossary

$f^{'} (x)$: first derivative or instantaneous rate
$f^{''} (x)$: second derivative
$\int$: integration or accumulation
$C$: arbitrary constant of integration

Student tools

Open graph studio Guided calculators