Cheatsheet

98. Cheatsheet

98.0.1. Limits

Epsilon-Delta

For every distance $𝜀$ around $𝐿$ , there’s a $𝛿$ -range around $𝑎$ that keeps $𝑓 (𝑥)$ within $𝜀$ of $𝐿$ .

lim_{𝑥 \to 𝑥_{0}} 𝑓 (𝑥) = 𝐿 ⟺ \forall 𝜀 > 0, \exists 𝛿 > 0 s.t. 0 < | 𝑥 - 𝑥_{0} | < 𝛿 \Rightarrow | 𝑓 (𝑥) - 𝐿 | < 𝜀

The limit of $𝑓 (𝑥)$ as $𝑥$ approaches $𝑥_{0}$ equals $𝐿$ if and only if, for every $𝜀 > 0$ , there exists a $𝛿 > 0$ such that, whenever $0 < | 𝑥 - 𝑥_{0} | < 𝛿$ , implies that $| 𝑓 (𝑥) - 𝐿 | < 𝜀$

Limit Type	Name	Quantifiers
$𝑥 \to 𝑥_{0}, 𝑓 (𝑥) \to 𝐿$	Epsilon-Delta	$\forall 𝜀, \exists 𝛿$
$𝑥 \to 𝑥_{0}, 𝑓 (𝑥) \to \infty$	M-Delta	$\forall 𝑀, \exists 𝛿$
$𝑥 \to \infty, 𝑓 (𝑥) \to 𝐿$	epsilon-N	$\forall 𝜀, \exists 𝑁$
$𝑥 \to \infty, 𝑓 (𝑥) \to \infty$	M-N	$\forall 𝑀, \exists 𝑁$

Finite $\to$ Finite (Epsilon-Delta)

lim_{𝑥 \to 𝑥_{0}} 𝑓 (𝑥) = 𝐿 ⟺ \forall 𝜀 > 0, \exists 𝛿 > 0 s.t. 0 < | 𝑥 - 𝑥_{0} | < 𝛿 \Rightarrow | 𝑓 (𝑥) - 𝐿 | < 𝜀

Finite $\to$ Infinity (M-Delta)

$+ \infty$

lim_{𝑥 \to 𝑥_{0}} 𝑓 (𝑥) = + \infty ⟺ \forall 𝑀 > 0, \exists 𝛿 > 0 s.t. 0 < | 𝑥 - 𝑥_{0} | < 𝛿 \Rightarrow 𝑓 (𝑥) > 𝑀

$- \infty$

lim_{𝑥 \to 𝑥_{0}} 𝑓 (𝑥) = - \infty ⟺ \forall 𝑀 > 0, \exists 𝛿 > 0 s.t. 0 < | 𝑥 - 𝑥_{0} | < 𝛿 \Rightarrow 𝑓 (𝑥) < - 𝑀

Infinity $\to$ Finite (Epsilon-N)

$+ \infty$

lim_{𝑥 \to + \infty} 𝑓 (𝑥) = 𝐿 ⟺ \forall 𝜀 > 0, \exists 𝑁 > 0 s.t. 𝑥 > 𝑁 \Rightarrow | 𝑓 (𝑥) - 𝐿 | < 𝜀

$- \infty$

lim_{𝑥 \to + \infty} 𝑓 (𝑥) = 𝐿 ⟺ \forall 𝜀 > 0, \exists 𝑁 > 0 s.t. 𝑥 < - 𝑁 \Rightarrow | 𝑓 (𝑥) - 𝐿 | < 𝜀

Infinity $\to$ Infinity (M-N)

lim_{𝑥 \to \pm \infty} 𝑓 (𝑥) = \pm \infty ⟺ \forall 𝑀 > 0, \exists 𝑁 > 0 s.t. 𝑥 < 𝑁 \Rightarrow 𝑓 (𝑥) > 𝑀

98.0.2. Derivatives

\frac{𝑑 𝑓}{𝑑 𝑥} (𝑥) = 𝑓^{'} (𝑥) = lim_{ℎ \to 0} \frac{𝑓 (𝑥 + ℎ) - 𝑓 (𝑥)}{ℎ}

Example

Let $𝑓 (𝑥) = 𝑥^{2}$

a. Find $𝑓^{'} (𝑥)$

𝑓^{'} (𝑥) = 2x

b. Prove a

\begin{aligned} 𝑓^{'} (𝑥) & = lim_{ℎ \to 0} \frac{𝑓 (𝑥 + ℎ) - 𝑓 (𝑥)}{(𝑥 + ℎ) - 𝑥} \\ = lim_{ℎ \to 0} \frac{𝑓 (𝑥 + ℎ) - 𝑓 (𝑥)}{ℎ} \\ = lim_{ℎ \to 0} \frac{{(𝑥 + ℎ)}^{2} - 𝑥^{2}}{ℎ} \\ = lim_{ℎ \to 0} \frac{𝑥^{2} + 2 𝑥 ℎ + ℎ^{2} - 𝑥^{2}}{ℎ} \\ = lim_{ℎ \to 0} \frac{ℎ (2 𝑥 + ℎ)}{ℎ} \\ = lim_{ℎ \to 0} (2 𝑥 + ℎ) \\ = lim_{ℎ \to 0} (2 𝑥 + 0) \\ = 2x \end{aligned}

c. Prove b

lim_{𝑥 \to 𝑥_{0}} 𝑓 (𝑥) = 𝐿 ⟺ \forall 𝜀 > 0, \exists 𝛿 > 0 s.t. 0 < | 𝑥 - 𝑥_{0} | < 𝛿 \Rightarrow | 𝑓 (𝑥) - 𝐿 | < 𝜀

p.f.:

Let $𝜀 > 0$

Choose $𝛿 = 𝜀$

Suppose $0 < | ℎ - 0 | < 𝛿$

Check

| \frac{{(𝑥 + ℎ)}^{2} - 𝑥^{2}}{ℎ} - 2 𝑥 |

\begin{aligned} = | \frac{𝑥^{2} + 2 𝑥 ℎ + ℎ^{2} - 𝑥^{2}}{ℎ} - 2 𝑥 | \\ = | \frac{ℎ (2 𝑥 + ℎ)}{ℎ} - 2 𝑥 | \\ = | 2 𝑥 + ℎ - 2 𝑥 | \\ = | ℎ | < 𝛿 = 𝜀 \end{aligned}

98.0.3. Integrals

98.0.4. Differential

If you have a function:

𝑦 = 𝑓 (𝑥)

Then the differential $d 𝑢$ is defined as:

d 𝑦 = 𝑓^{'} (𝑥) \cdot d 𝑥

This means:

$𝑓^{'} (𝑥) = \frac{d 𝑦}{d 𝑥}$ is the derivative
$d 𝑥$ is a small change in $𝑥$
$d 𝑦$ is the corresponding small change in $𝑦$

So:

d 𝑦 = \frac{d 𝑦}{d 𝑥} \cdot d 𝑥

Example

Let’s say“

𝑦 = 𝑥^{2}

Then:

\frac{d 𝑦}{d 𝑥} = 2 𝑥

\frac{𝑑}{d 𝑥} (\int 𝑓 (𝑥) d 𝑥) = 𝑓 (𝑥)

\int (\frac{𝑑}{d 𝑥} 𝑓 (𝑥)) d 𝑥 = 𝑓 (𝑥)

Operation	Notation	Input	Output	Meaning
Derivative	$\frac{𝑑 𝑓}{𝑑 𝑥}$	$\begin{matrix} Function \\ 𝑓 (𝑥) \end{matrix}$	$\begin{matrix} Function \\ 𝑓^{'} (𝑥) \end{matrix}$	Slope/Rate of change at each $𝑥$
Indefinite Integral	$\int 𝑓 (𝑥) 𝑑 𝑥$	$\begin{matrix} Function \\ 𝑓 (𝑥) \end{matrix}$	$\begin{matrix} Family of functions \\ 𝐹 (𝑥) + 𝐶 \end{matrix}$	Function whose slope is $𝑓 (𝑥)$
Definite Integral	$\int_{𝑎}^{𝑏} 𝑓 (𝑥) 𝑑 𝑥$	$\begin{matrix} Function \\ 𝑓 (𝑥) \\ Bounds \\ [𝑎, 𝑏] \end{matrix}$	$Number$	Total signed area between $𝑎$ and $𝑏$

Rule	$\frac{𝑑}{d 𝑥}$ Rule	$\frac{𝑑}{d 𝑥}$ Example	$\int$ Rule	$\int$ Example
Constant	$\frac{𝑑}{d 𝑥} [𝑐] = 0$	$\frac{𝑑}{d 𝑥} [7] = 0$	$\int 𝑐 d 𝑥 = 𝑐 𝑥 + 𝐶$	$\int 7 d 𝑥 = 7 𝑥 + 𝐶$
Power	$\frac{𝑑}{d 𝑥} [𝑥^{𝑛}] = 𝑛 𝑥^{𝑛 - 1}$	$\frac{𝑑}{d 𝑥} [𝑥^{4}] = 4 𝑥^{3}$	$\begin{matrix} \int 𝑥^{𝑛} d 𝑥 = \frac{𝑥^{𝑛 + 1}}{𝑛 + 1} + 𝐶 \\ (𝑛 \neq - 1) \end{matrix}$	$\int 𝑥^{3} d 𝑥 = \frac{𝑥^{4}}{4} + 𝐶$
Constant Multiple	$\frac{𝑑}{d 𝑥} [𝑐 𝑓] = 𝑐 𝑓^{'}$	$\begin{aligned} \frac{𝑑}{d 𝑥} [3 𝑥^{2}] & = 3 \cdot 2 𝑥 \\ = 6 𝑥 \end{aligned}$	$\int 𝑐 𝑓 d 𝑥 = 𝑐 \int 𝑓 d 𝑥$	$\begin{aligned} \int 3 𝑥^{2} d 𝑥 & = 3 \cdot \frac{𝑥^{3}}{3} \\ = 𝑥^{3} + 𝐶 \end{aligned}$
Sum	$\frac{𝑑}{𝑑 𝑥} [𝑓 + 𝑔] = 𝑓^{'} + 𝑔^{'}$	$\begin{aligned} \frac{𝑑}{d 𝑥} (𝑥^{2} + 𝑥) \\ = \frac{𝑑}{d 𝑥} (𝑥^{2}) + \frac{𝑑}{d 𝑥} (𝑥) \\ = 2 𝑥 + 1 \end{aligned}$	$\begin{matrix} \int (𝑓 + 𝑔) d 𝑥 \\ = \int 𝑓 d 𝑥 + \int 𝑔 d 𝑥 \end{matrix}$	$\begin{aligned} \int (𝑥^{2} + 𝑥) d 𝑥 \\ = \int 𝑥^{2} d 𝑥 + \int 𝑥 d 𝑥 \\ = \frac{𝑥^{3}}{3} + \frac{3^{2}}{2} + 𝐶 \end{aligned}$
Difference	$\frac{𝑑}{𝑑 𝑥} [𝑓 - 𝑔] = 𝑓^{'} - 𝑔^{'}$	$\begin{aligned} \frac{𝑑}{d 𝑥} (𝑥^{2} - 𝑥) \\ = \frac{𝑑}{d 𝑥} (𝑥^{2}) - \frac{𝑑}{d 𝑥} (𝑥) \\ = 2 𝑥 - 1 \end{aligned}$	$\begin{matrix} \int (𝑓 - 𝑔) d 𝑥 \\ = \int 𝑓 d 𝑥 - \int 𝑔 d 𝑥 \end{matrix}$	$\begin{aligned} \int (𝑥^{2} - 𝑥) d 𝑥 \\ = \int 𝑥^{2} d 𝑥 - \int 𝑥 d 𝑥 \\ = \frac{𝑥^{3}}{3} - \frac{𝑥^{2}}{2} + 𝐶 \end{aligned}$
Product	$\frac{𝑑}{𝑑 𝑥} [𝑓 𝑔] = 𝑓^{'} 𝑔 + 𝑓 𝑔^{'}$	$\begin{aligned} \frac{𝑑}{d 𝑥} [𝑥 sin 𝑥] \\ = 1 \cdot sin 𝑥 + 𝑥 \cdot cos 𝑥 \end{aligned}$	Integration by Parts $\int 𝑢 d 𝑣 = 𝑢 𝑣 - \int 𝑣 d 𝑢$
Quotient	$\frac{𝑑}{𝑑 𝑥} [\frac{𝑓}{𝑔}] = \frac{𝑓^{'} 𝑔 - 𝑓 𝑔^{'}}{𝑔^{2}}$	$\begin{aligned} \frac{𝑑}{d 𝑥} [\frac{𝑥^{2}}{𝑥 + 1}] \\ = \frac{2 𝑥 (𝑥 + 1) - 𝑥^{2} (1)}{{(𝑥 + 1)}^{2}} \\ = \frac{𝑥^{2} + 2 𝑥}{{(𝑥 + 1)}^{2}} \end{aligned}$	Algebraic Manipulation / Substitution
Chain	$\begin{aligned} \frac{𝑑}{𝑑 𝑥} [𝑓 (𝑔 (𝑥))] \\ = 𝑓^{'} (𝑔 (𝑥)) \cdot 𝑔^{'} (𝑥) \end{aligned}$	$\begin{aligned} \frac{𝑑}{d 𝑥} [sin (𝑥^{2})] \\ = cos (𝑥^{2}) \cdot 2 𝑥 \end{aligned}$	Integration by Substitution $\begin{matrix} \int 𝑓 (𝑔 (𝑥)) 𝑔^{'} (𝑥) d 𝑥 \\ = \int 𝑓 (𝑢) d 𝑢 \end{matrix}$
Exponential	$\frac{𝑑}{d 𝑥} [𝑒^{𝑥}] = 𝑒^{𝑥}$	$\frac{𝑑}{d 𝑥} [𝑒^{𝑥}] = 𝑒^{𝑥}$	$\int 𝑒^{𝑥} d 𝑥 = 𝑒^{𝑥} + 𝐶$	$\int 𝑒^{𝑥} d 𝑥 = 𝑒^{𝑥} + 𝐶$
	$\frac{𝑑}{𝑑 𝑥} [𝑎^{𝑥}] = 𝑎^{𝑥} ln (𝑎)$	$\frac{𝑑}{d 𝑥} [2^{𝑥}] = 2^{𝑥} ln 2$	$\int 𝑎^{𝑥} 𝑑 𝑥 = \frac{𝑎^{𝑥}}{ln 𝑎} + 𝐶$	$\int 2^{𝑥} d 𝑥 = \frac{2^{𝑥}}{ln 2} + 𝐶$
Logarithmic	$\frac{𝑑}{𝑑 𝑥} [ln (𝑥)] = \frac{1}{𝑥}$	$\frac{𝑑}{𝑑 𝑥} [ln (𝑥)] = \frac{1}{𝑥}$	$\begin{matrix} \int ln (𝑥) d 𝑥 \\ = 𝑥 ln 𝑥 - 𝑥 + 𝐶 \end{matrix}$	$\begin{matrix} \int ln (𝑥) d 𝑥 \\ = 𝑥 ln 𝑥 - 𝑥 + 𝐶 \end{matrix}$
	$\frac{𝑑}{𝑑 𝑥} [{log}_{𝑎} (𝑥)] = \frac{1}{𝑥 ln (𝑎)}$	$\frac{𝑑}{d 𝑥} {log}_{2} (𝑥) = \frac{1}{𝑥 ln (2)}$	$\begin{matrix} \int {log}_{𝑎} (𝑥) d 𝑥 = \\ = \frac{𝑥 ln 𝑥}{ln 𝑎} - \frac{𝑥}{ln 𝑎} + 𝐶 \end{matrix}$	$\begin{matrix} \int {log}_{10} (𝑥) d 𝑥 \\ = \frac{𝑥 ln 𝑥}{ln 10} - \frac{𝑥}{ln 10} + 𝐶 \end{matrix}$
Sin	$\frac{𝑑}{𝑑 𝑥} [sin (𝑥)] = cos (𝑥)$	$\frac{𝑑}{𝑑 𝑥} [sin (𝑥)] = cos (𝑥)$	$\begin{matrix} \int sin (𝑥) d 𝑥 \\ = - cos 𝑥 + 𝐶 \end{matrix}$	$\begin{matrix} \int sin (𝑥) d 𝑥 \\ = - cos 𝑥 + 𝐶 \end{matrix}$
Cos	$\frac{𝑑}{𝑑 𝑥} [cos (𝑥)] = - sin (𝑥)$	$\frac{𝑑}{𝑑 𝑥} [cos (𝑥)] = - sin (𝑥)$	$\begin{matrix} \int cos (𝑥) d 𝑥 \\ = sin 𝑥 + 𝐶 \end{matrix}$	$\begin{matrix} \int cos (𝑥) d 𝑥 \\ = sin 𝑥 + 𝐶 \end{matrix}$
Tan	$\begin{matrix} \frac{𝑑}{𝑑 𝑥} [tan (𝑥)] \\ = {sec}^{2} (𝑥) \end{matrix}$	$\begin{matrix} \frac{𝑑}{𝑑 𝑥} [tan (𝑥)] \\ = {sec}^{2} (𝑥) \end{matrix}$	$\begin{matrix} \int tan (𝑥) d 𝑥 \\ = - ln \| cos 𝑥 \| + 𝐶 \end{matrix}$	$\begin{matrix} \int tan (𝑥) d 𝑥 \\ = - ln \| cos 𝑥 \| + 𝐶 \end{matrix}$

98.0.5. Product Rule $⟶$ Integration by Parts

\int 𝑢 d 𝑣 = 𝑢 𝑣 - \int 𝑣 d 𝑢

Example

Given a function:

𝑓 (𝑥) = 𝑥 \cdot 𝑒^{𝑥}

Integrate (by parts):

\int 𝑥 \cdot 𝑒^{𝑥} d 𝑥

Step 1: Choose $𝑢$ and $d 𝑣$

We choose:

$𝑢 = 𝑥$ (easy to differentiate)
$d 𝑣 = 𝑒^{𝑥}$ (easy to integrate)

Step 2: Compute $d 𝑢$ and $𝑣$

$𝑢 = 𝑥 ⟶ d 𝑢 = d 𝑥$
$d 𝑣 = 𝑒^{𝑥} d 𝑥 ⟶ 𝑣 = \int 𝑒^{𝑥} d 𝑥 = 𝑒^{𝑥}$

Step 3: Plug into formula

\begin{aligned} \int 𝑥 \cdot 𝑒^{𝑥} d 𝑥 & = 𝑢 \cdot 𝑣 - \int 𝑣 \cdot d 𝑢 \\ = 𝑥 \cdot 𝑒^{𝑥} d 𝑥 - \int 𝑒^{𝑥} \cdot d 𝑥 \end{aligned}

Step 4: Compute the remaining integral:

\int 𝑒^{𝑥} d 𝑥 = 𝑒^{𝑥}

Step 5: Finish the expression:

\begin{aligned} \int 𝑥 \cdot 𝑒^{𝑥} & = 𝑥 \cdot 𝑒^{𝑥} - 𝑒^{𝑥} + 𝐶 \\ = 𝑒^{𝑥} (𝑥 - 1) + 𝐶 \end{aligned}

98.0.6. Chain Rule $⟶$ $𝑢$ -Substitution

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

98. Cheatsheet

98.0.1. Limits

98.0.2. Derivatives

98.0.3. Integrals

98.0.4. Differential

98.0.5. Product Rule ⟶ Integration by Parts

98.0.6. Chain Rule ⟶ 𝑢-Substitution

98.0.7. Quotient Rule ⟶ Algebraic Manipulation / Substitution

98.0.5. Product Rule $⟶$ Integration by Parts

98.0.6. Chain Rule $⟶$ $𝑢$ -Substitution

98.0.7. Quotient Rule $⟶$ Algebraic Manipulation / Substitution