Derivative Formulas with Rules, Conditions and PDF

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Single terms

Differentiate constants and powers before combining longer expressions.

$\frac{d}{dx}c=0$: Constant rulec is constant with respect to x.
Use this rule for a term that does not change as x changes.
$\frac{d}{dx}x^n=nx^{n-1}$: Power rulen is constant with respect to x. Work at points where the real-valued power is defined and differentiable.
Use this rule to differentiate a power of the variable.
$\frac{d}{dx}[f(x)+g(x)]=f'(x)+g'(x)$: Sum rulef and g are differentiable at the point being evaluated.
Use this rule to differentiate a sum term by term.

Products, quotients, and compositions

Choose a structural rule when functions multiply, divide, or sit inside one another.

$\frac{d}{dx}[f(x)g(x)]=f'(x)g(x)+f(x)g'(x)$: Product rulef and g are differentiable at the point being evaluated.
Use this rule when two differentiable functions are multiplied.
$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right]=\frac{f'(x)g(x)-f(x)g'(x)}{[g(x)]^2}$: Quotient rulef and g are differentiable at the point being evaluated. g of x is not zero at that point.
Use this rule when one differentiable function is divided by another.
$\frac{d}{dx}f(g(x))=f'(g(x))g'(x)$: Chain ruleg is differentiable at x. f is differentiable at g of x.
Use this rule when one differentiable function is composed inside another.

Exponential, logarithmic, and trigonometric functions

Use these base derivatives directly or inside the chain rule.

$\frac{d}{dx}e^x=e^x$: Natural exponential derivativex is real for the real-valued form shown.
Use this rule for the natural exponential function; combine it with the chain rule when the exponent is not x.
$\frac{d}{dx}\ln x=\frac{1}{x}$: Natural logarithm derivativex is greater than zero.
Use this rule to differentiate the natural logarithm on its real domain.
$\frac{d}{dx}\sin x=\cos x$: Sine derivativeAngles are measured in radians.
Use this rule to differentiate sine; combine it with the chain rule for sine of another function.

Notation

Symbols used on this sheet

Symbol	Meaning	Usage note
$x$	Differentiation variable	Other symbols are constants only when the problem states or implies that role.
$f,g$	Functions	Use each formula only where the required derivatives exist.
$f'(x)$	Derivative of f at x	It represents an instantaneous rate or tangent slope when the interpretation applies.
$n$	Constant exponent	Real powers can require domain restrictions even when the symbolic rule looks familiar.
$g(x)\ne0$	Nonzero denominator condition	The quotient rule is stated only where the original denominator is nonzero.
$\ln x$	Natural logarithm	The real logarithm formula shown requires positive x.

Use the reference

Short applications

These examples show when to choose a formula; detailed instruction belongs in the linked guide or calculator.

A product with two changing factors

The polynomial and exponential both depend on x, so the product rule retains each factor once.

\frac{d}{dx}(x^2e^x)

2xe^x+x^2e^x=e^x(x^2+2x)

Condition check: A centered difference quotient agrees at regular sample points, and expanding the factored result returns both product-rule terms.

A logarithm containing a quadratic

The logarithm is the outer function and x squared plus 1 is the inner function.

\frac{d}{dx}\ln(x^2+1)

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

Condition check: The denominator is positive for every real x, and direct chain-rule differentiation returns the displayed result.

Use conditions, not memory alone

Common formula confusions

Choosing by notation alone

Seeing parentheses and applying the chain rule even when two functions are multiplied.

Fix: Identify addition, multiplication, division, or composition at the outermost level before selecting a formula.

Dropping an unchanged factor

Writing f prime plus g prime for the derivative of a product.

Fix: Use f prime times g plus f times g prime, keeping the untouched factor in each term.

Reversing the quotient numerator

Switching the subtraction order in the quotient rule.

Fix: Write denominator times numerator derivative minus numerator times denominator derivative before simplifying.

Ignoring the original domain

Accepting a simplified derivative at a point where the original logarithm, radical, or denominator is undefined.

Fix: Carry the original domain beside the derivative and check boundary points separately.

Common questions

Derivative formulas FAQ

Which derivative formulas are included?

The focused sheet includes constant, power, sum, product, quotient, chain, natural exponential, natural logarithm, and sine derivative formulas, each with its conditions.

How do I decide between the product rule and chain rule?

Use the product rule when two functions are multiplied. Use the chain rule when one function is evaluated inside another, such as sine of x squared.

Why can two correct derivative answers look different?

Factoring, expanding, trigonometric identities, and algebraic simplification can produce equivalent forms. Compare their difference symbolically and keep the shared domain in view.

Does the PDF contain the same formulas as this page?

Yes. The downloadable derivative sheet and the HTML reference are generated from the same structured formula entries and conditions.

Sources and curriculum alignment

This page follows standard introductory mathematics notation and learning sequences. Use these references to continue with a complete course treatment.

3.3 Differentiation Rules, OpenStax Calculus Volume 1
3.6 The Chain Rule, OpenStax Calculus Volume 1

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