What does this Calculus I: Derivatives cheat sheet cover?

It has 10 sections: Derivative Basics, Basic Rules, Trig / Exp / Log, Inverse Trig, Implicit / Related, Linearization, L'Hôpital, Critical Tests, Optimization and Mean Value. Together they cover the topics a typical Calculus I: Derivatives exam tests, including limit definition of the derivative, power, product, quotient rules, chain rule and derivatives of trig, exponential, log functions.

Is the derivative rules cheat sheet free?

The first part is free to read on this page, with no account needed. The full sheet opens in the app with Unlimited ($4.99 a month or $19 for a semester), where you can edit it, print it or save it as a PDF. It is one US Letter page and prints cleanly in black and white or colour.

Can I bring this Calculus I: Derivatives cheat sheet to my exam?

Only if your course allows a note sheet, so check the syllabus. This one is a single US Letter page. If your exam allows a different size, such as a 3x5 or 4x6 index card or one page front and back, Cheat Sheet AI makes one at exactly that size from your own lecture notes.

The sheet was written by our engine from a standard course outline, then a second model fact-checked every formula and statement against it; it corrected 2 statements before publishing. Notation and emphasis vary between courses, so check it against your own lecture notes.

Can I make a Calculus I: Derivatives sheet from my own notes?

Yes. Upload your lecture slides, notes or study guide and Cheat Sheet AI writes a sheet in your course's notation, sized to exactly the space your exam allows. Your first sheets are free.

MathCalculus I: Derivatives

Derivative Rules Cheat Sheet — 1-page PDF

Limit definition, rules, trig/exp/log, implicit diff, applications, tests. The whole course on one dense, printable page for your midterm or final, with every formula typeset and worked examples where they help.

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For a specific exam: Calculus I Midterm 1 study sheet, Calculus I Final exam study sheet.

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10 sections12 topics
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Preview of the one-page Calculus I: Derivatives cheat sheet: Limit definition, rules, trig/exp/log, implicit diff, applications, tests — the whole course, one page

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What’s on it

The topics a typical Calculus I: Derivatives exam tests, all on the one page.

Limit definition of the derivative
Power, product, quotient rules
Chain rule
Derivatives of trig, exponential, log functions
Inverse trig derivatives
Implicit differentiation
Related rates
Linearization and differentials
L'Hôpital's rule
Critical points, first and second derivative tests
Optimization
Mean value theorem

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The free part of the sheet, section by section. Unlock the full sheet to read the rest, edit any block, and print it or save it as a PDF.

01Derivative Basics

Derivative at $a$: $f^{'} (a) = lim_{h \to 0} \frac{f ( a + h ) - f ( a )}{h}$ if the limit exists. Slope of tangent line at $x = a$ ; instantaneous rate of change.

Equivalent form

f^{'} (a) = lim_{x \to a} \frac{f ( x ) - f ( a )}{x - a}

Same definition; use whichever variable is given.

Derivative function

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

Defines the derivative as a function of $x$ .

Differentiable: $f$ differentiable at $a$ iff $f^{'} (a)$ exists. Differentiable $\Rightarrow$ continuous at $a$ .

Tangent line

y - f (a) = f^{'} (a) (x - a)

Line tangent to $y = f (x)$ at $x = a$ .

Normal line

y - f (a) = - \frac{1}{f ^{'} ( a )} (x - a)

Requires $f^{'} (a) \neq = 0$ .

Derivative meaning

Expression	Meaning
$f^{'} (a) > 0$	increasing at $a$
$f^{'} (a) < 0$	decreasing at $a$
$f^{'} (a) = 0$	horizontal tangent / critical candidate

02Trig / Exp / Log

Exponential

\frac{d}{d x} (e^{x}) = e^{x}, \frac{d}{d x} (a^{x}) = a^{x} ln a

$a > 0$ , $a \neq = 1$ .

Logarithm

\frac{d}{d x} (ln x) = \frac{1}{x}, \frac{d}{d x} (lo g_{a} x) = \frac{1}{x l n a}

For $x > 0$ ; $a > 0$ , $a \neq = 1$ .

Trig + chain

\frac{d}{d x} [sin (g (x))] = cos (g (x)) g^{'} (x)

Same pattern for all basic trig/exponential/log functions.

Common patterns

Function	Derivative
$e^{g (x)}$	$e^{g (x)} g^{'} (x)$
$ln (g (x))$	$g^{'} (x) / g (x)$
$a^{g (x)}$	$a^{g (x)} ln a g^{'} (x)$

Ex.Differentiate $y = ln (sin x)$ .

Chain rule: $y^{'} = \frac{1}{s i n x} cos x = cot x$ .

Watch out:For $ln (g (x))$ , derivative is $g^{'} (x) / g (x)$ , not $1/ ln (g (x))$ .

03Inverse Trig

Inverse trig derivatives

(arcsin x)^{'} = \frac{1}{1 - x ^{2}}, (arccos x)^{'} = - \frac{1}{1 - x ^{2}}, (arctan x)^{'} = \frac{1}{1 + x ^{2}}

Also $(arccot x)^{'} = - \frac{1}{1 + x ^{2}}$ if used by course.

05Linearization

Linearization

L (x) = f (a) + f^{'} (a) (x - a)

Best linear approximation near $x = a$ .

Differential

d y = f^{'} (x) d x

At $x = a$ : $Δ y \approx d y = f^{'} (a) d x$ .

Approximation

f (a + Δ x) \approx f (a) + f^{'} (a) Δ x

For small $Δ x$ .

Error idea

Δ y = f^{'} (a) Δ x + small error

Use to estimate change in $f$ .

Ex.Linearize $f (x) = x$ at $a = 4$ . Estimate $4.1$ .

$f (4) = 2$ , $f^{'} (x) = 1/ (2 x)$ , so $f^{'} (4) = 1/4$ . Thus $L (x) = 2 + \frac{1}{4} (x - 4)$ . Then $4.1 \approx 2 + \frac{1}{4} (0.1) = 2.025$ .

06Critical Tests

Concavity

$f^{''} (x) > 0 \Rightarrow$ concave up $, f^{''} (x) < 0 \Rightarrow$ concave down

Inflection point where concavity changes.

Test choices

Need	Use
sign change of $f^{'}$	1st derivative test
easy $f^{''}$ at crit. point	2nd derivative test
global extrema on closed interval	endpoint + critical numbers

Watch out:For absolute max/min on $[a, b]$ , check endpoints too; local extrema can miss the global answer.

07Optimization

Optimization setup: Max/min a quantity $Q$ by writing $Q$ as a function of one variable using the constraint, then analyze critical numbers and endpoints if relevant.

Define variable(s) and quantity $Q$
Write constraint equation
Reduce to $Q (x)$ in one variable
Find $Q^{'} (x) = 0$ and critical numbers
Use tests + endpoints to decide extrema

Closed interval

Absolute extrema on $[a, b]$ : compare $f (a), f (b),$ and critical values

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Also on the full Calculus I: Derivatives sheet

10 more sections, with every definition, formula and worked example. Unlimited unlocks every full sheet in the library to edit, print or save as a PDF, and makes unlimited sheets from your own notes.

Derivative Basics

Basic Rules

Constant
Power
Constant multiple
Sum/difference
Product
Quotient
Chain rule

Trig / Exp / Log

Trig derivatives

Inverse Trig

More inverses
Chain form

Implicit / Related

Implicit differentiation
Key idea
Product in implicit
Related rates
Related-rates steps

Linearization

L'Hôpital

L'Hôpital
Indeterminate forms
0·∞ trick

Critical Tests

Critical number
Local max/min
First derivative test
Second derivative test

Optimization

Mean Value

Rolle's Theorem
MVT
Average rate of change
MVT checklist

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About the Calculus I: Derivatives sheet and making your own.

What does this Calculus I: Derivatives cheat sheet cover?: It has 10 sections: Derivative Basics, Basic Rules, Trig / Exp / Log, Inverse Trig, Implicit / Related, Linearization, L'Hôpital, Critical Tests, Optimization and Mean Value. Together they cover the topics a typical Calculus I: Derivatives exam tests, including limit definition of the derivative, power, product, quotient rules, chain rule and derivatives of trig, exponential, log functions.
Is the derivative rules cheat sheet free?: The first part is free to read on this page, with no account needed. The full sheet opens in the app with Unlimited ($4.99 a month or $19 for a semester), where you can edit it, print it or save it as a PDF. It is one US Letter page and prints cleanly in black and white or colour.
Can I bring this Calculus I: Derivatives cheat sheet to my exam?: Only if your course allows a note sheet, so check the syllabus. This one is a single US Letter page. If your exam allows a different size, such as a 3x5 or 4x6 index card or one page front and back, Cheat Sheet AI makes one at exactly that size from your own lecture notes.
How accurate is it?: The sheet was written by our engine from a standard course outline, then a second model fact-checked every formula and statement against it; it corrected 2 statements before publishing. Notation and emphasis vary between courses, so check it against your own lecture notes.
Can I make a Calculus I: Derivatives sheet from my own notes?: Yes. Upload your lecture slides, notes or study guide and Cheat Sheet AI writes a sheet in your course's notation, sized to exactly the space your exam allows. Your first sheets are free.

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Derivative Rules Cheat Sheet — 1-page PDF

What’s on it

The free part, as text

01Derivative Basics

02Trig / Exp / Log

03Inverse Trig

04Implicit / Related

05Linearization

06Critical Tests

07Optimization

Also on the full Calculus I: Derivatives sheet

Derivative Basics

Basic Rules

Trig / Exp / Log

Inverse Trig

Implicit / Related

Linearization

L'Hôpital

Critical Tests

Optimization

Mean Value

Questions

More sheets

More Math

Other subjects

Your course, your notes, one page.