∬Differential Calculus

Common Derivative Formulas

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Why This Matters

Derivative formulas are the essential toolkit you'll use throughout calculus—they're not just rules to memorize but patterns that reveal how functions change. Every AP Calculus problem involving rates of change, optimization, curve analysis, or related rates depends on your ability to quickly and accurately differentiate functions. You're being tested on your fluency with these formulas and, more importantly, your judgment about which rule applies when.

The key insight is that all derivative formulas stem from a few core principles: linearity (derivatives distribute over addition), the limit definition (which generates the power and exponential rules), and function composition (which demands the chain rule). Don't just memorize formulas—know what type of function structure each rule handles. When you see a product, think Product Rule. When you see a function inside another function, think Chain Rule. This pattern recognition is what separates students who struggle from those who breeze through the exam.

Foundational Rules: Building Blocks for Everything

These rules form the backbone of differentiation. Master them first, because every other technique builds on these principles. Linearity—the idea that differentiation is a linear operation—means you can break complex functions into simpler pieces.

Constant Rule

$\frac{d}{dx}(c) = 0$ —constants have no rate of change, so their derivative is always zero
Graphical interpretation: a horizontal line has zero slope everywhere
Strategic use: lets you ignore constant terms when differentiating, simplifying your work significantly

Power Rule

$\frac{d}{dx}(x^n) = nx^{n-1}$ —bring down the exponent, then reduce it by one
Works for all real exponents: integers, fractions, and negatives (e.g., $x^{1/2}$ , $x^{-3}$ )
Most frequently tested rule: appears in nearly every derivative problem, often combined with Chain Rule

Sum/Difference Rule

$\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$ —differentiate each term separately
Linearity property: allows you to break polynomials into individual power functions
No product concerns: only applies when functions are added or subtracted, not multiplied

Compare: Constant Rule vs. Power Rule—both handle simple terms, but Power Rule applies when the variable has an exponent while Constant Rule handles pure numbers. Remember: $x^0 = 1$ , so constants are technically $x^0$ with derivative $0 \cdot x^{-1} = 0$ .

Combination Rules: When Functions Interact

When two functions are multiplied, divided, or nested inside each other, you need specialized rules. These are where most students make errors—memorize the formulas precisely and practice identifying which structure you're dealing with.

Product Rule

$\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$ —"derivative of first times second, plus first times derivative of second"
Common mnemonic: "first d-second plus second d-first" helps recall the pattern
FRQ essential: frequently appears in related rates and applications where quantities multiply

Quotient Rule

$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$ —"low d-high minus high d-low, over low squared"
Watch the subtraction order: numerator derivative comes first; reversing gives the wrong sign
Alternative approach: rewrite as $f(x) \cdot [g(x)]^{-1}$ and use Product Rule with Chain Rule

Chain Rule

$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$ —differentiate outside, keep inside, then multiply by derivative of inside
Identifies composite functions: whenever you see "something inside something" (like $\sin(3x)$ or $(x^2+1)^5$ )
Most commonly missed rule: if your answer is wrong, check whether you forgot to multiply by the inner derivative

Compare: Product Rule vs. Chain Rule—Product Rule handles $f(x) \cdot g(x)$ (two separate functions multiplied), while Chain Rule handles $f(g(x))$ (one function inside another). Confusing these is a top exam error. Ask yourself: "Are these multiplied together, or is one inside the other?"

Exponential and Logarithmic Rules: Growth and Decay

These functions model real-world phenomena like population growth, radioactive decay, and compound interest. Their derivatives have elegant forms that make them favorites on the AP exam.

Natural Exponential Function

$\frac{d}{dx}(e^x) = e^x$ —the only function that is its own derivative
With Chain Rule: $\frac{d}{dx}(e^{u}) = e^{u} \cdot u'$ where $u$ is any function of $x$
Why $e$ is special: this self-derivative property is precisely what defines the number $e \approx 2.718$

Natural Logarithm

$\frac{d}{dx}(\ln x) = \frac{1}{x}$ —valid only for $x > 0$ (domain restriction)
Inverse relationship: since $\ln x$ and $e^x$ are inverses, their derivatives are reciprocally related
With Chain Rule: $\frac{d}{dx}(\ln u) = \frac{u'}{u}$ , extremely useful for logarithmic differentiation

Compare: $e^x$ vs. $\ln x$ —they're inverse functions, and their derivatives reflect this: $e^x$ stays as $e^x$ , while $\ln x$ becomes $\frac{1}{x}$ . On FRQs involving growth models, you'll often need both.

Trigonometric Rules: Periodic Functions

Trig derivatives follow patterns—sine and cosine cycle into each other, while tangent and its cousins involve squared secants and cosecants. Watch the negative signs carefully.

Sine and Cosine

$\frac{d}{dx}(\sin x) = \cos x$ —sine's rate of change equals cosine
$\frac{d}{dx}(\cos x) = -\sin x$ —note the negative sign; cosine decreases when sine is positive
Cyclic pattern: differentiating four times returns you to the original function

Tangent

$\frac{d}{dx}(\tan x) = \sec^2 x$ —derived using Quotient Rule on $\frac{\sin x}{\cos x}$
Always positive: $\sec^2 x \geq 1$ , so tangent is always increasing on its domain
Related results: $\frac{d}{dx}(\cot x) = -\csc^2 x$ , $\frac{d}{dx}(\sec x) = \sec x \tan x$ , $\frac{d}{dx}(\csc x) = -\csc x \cot x$

Compare: $\sin x$ vs. $\cos x$ derivatives—they cycle into each other with a sign change. The pattern is: $\sin \to \cos \to -\sin \to -\cos \to \sin$ . Knowing this cycle helps with higher-order derivatives.

Inverse Trigonometric Rules: Working Backwards

These derivatives look intimidating but follow predictable patterns. They appear frequently in integration (as antiderivatives) and in problems involving angles and geometry.

Arcsine and Arccosine

$\frac{d}{dx}(\arcsin x) = \frac{1}{\sqrt{1-x^2}}$ —valid for $-1 < x < 1$
$\frac{d}{dx}(\arccos x) = -\frac{1}{\sqrt{1-x^2}}$ —same magnitude, opposite sign
Domain matters: the square root restricts inputs to $|x| < 1$

Arctangent

$\frac{d}{dx}(\arctan x) = \frac{1}{1+x^2}$ —valid for all real $x$
No domain restrictions: unlike arcsine/arccosine, works everywhere
Integration connection: recognizing this form helps identify $\arctan$ as an antiderivative

Compare: $\arcsin x$ vs. $\arctan x$ —both are inverse trig derivatives, but $\arcsin$ has restricted domain ( $|x| < 1$ ) while $\arctan$ works for all real numbers. If an FRQ gives you $\frac{1}{1+x^2}$ , think arctangent immediately.

Quick Reference Table

Concept	Best Examples
Linearity/Basic Rules	Constant Rule, Power Rule, Sum/Difference Rule
Product Structures	Product Rule
Quotient Structures	Quotient Rule
Composite Functions	Chain Rule (used with all other rules)
Exponential/Logarithmic	$e^x$ , $\ln x$
Basic Trigonometric	$\sin x$ , $\cos x$ , $\tan x$
Inverse Trigonometric	$\arcsin x$ , $\arccos x$ , $\arctan x$
Self-Derivative Property	$e^x$ (only function equal to its own derivative)

Self-Check Questions

Which two rules both require you to differentiate multiple functions and combine results, but differ in whether you add or subtract terms?
You need to differentiate $f(x) = e^{3x^2}$ . Which rules must you combine, and in what order do you apply them?
Compare and contrast the derivatives of $\arcsin x$ and $\arccos x$ —what do they share, and what's different?
A student differentiates $(x^2 + 1)(x^3 - x)$ and gets $2x \cdot 3x^2$ . What rule did they incorrectly apply, and what should they have used?
If an FRQ asks you to find where $f(x) = \tan x$ is increasing, what does the derivative $\sec^2 x$ tell you about the answer?