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Calculus formula sheet

Calculus formulas

Scan the core rules, check when each one applies, and download the same structured reference as a PDF.

This calculus formula sheet groups frequently used derivative, integral, and limit rules. Each formula includes a condition because remembering a pattern without its domain or hypotheses is a common source of errors.

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Derivative rules

Differentiate basic functions, sums, products, quotients, and compositions.

ddxc=0\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.

ddxxn=nxn−1\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.

ddx[f(x)+g(x)]=f′(x)+g′(x)\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.

ddx[f(x)g(x)]=f′(x)g(x)+f(x)g′(x)\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.

ddx[f(x)g(x)]=f′(x)g(x)−f(x)g′(x)[g(x)]2\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.

ddxf(g(x))=f′(g(x))g′(x)\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.

ddxex=ex\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.

ddxln⁡x=1x\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.

ddxsin⁡x=cos⁡x\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.

Integral rules

Recognize common antiderivatives and evaluate definite integrals.

∫xn dx=xn+1n+1+C\int x^n\,dx=\frac{x^{n+1}}{n+1}+C
Power antiderivativen is constant with respect to x. n is not negative one. Use an interval where the real-valued power is defined.

Use this rule to integrate a power of x other than x to the negative one.

∫ex dx=ex+C\int e^x\,dx=e^x+C
Natural exponential antiderivativeC is an arbitrary constant of integration.

Use this rule to integrate the natural exponential function.

∫1x dx=ln⁡∣x∣+C\int \frac{1}{x}\,dx=\ln|x|+C
Reciprocal antiderivativex is not zero. Choose an interval entirely on one side of zero. C is an arbitrary constant of integration.

Use this rule for the reciprocal function on an interval that does not cross zero.

∫u dv=uv−∫v du\int u\,dv=uv-\int v\,du
Integration by partsu and v are differentiable on the interval of interest. The displayed integrals exist on that interval.

Use this identity when differentiating one factor simplifies it and the other factor can be integrated.

∫abf(x) dx=F(b)−F(a)\int_a^b f(x)\,dx=F(b)-F(a)
Definite integral evaluationf is continuous on the closed interval from a to b. F is an antiderivative of f on that interval.

Use this result to evaluate a definite integral from a verified antiderivative.

Limit rules

Combine known limits and use the standard sine limit with the correct conditions.

lim⁡x→a[f(x)+g(x)]=L+M\lim_{x\to a}[f(x)+g(x)]=L+M
Limit sum lawThe limit of f of x as x approaches a equals L. The limit of g of x as x approaches a equals M. L and M are finite real numbers for the form shown.

Use this law to split the limit of a sum when both component limits exist.

lim⁡x→a[f(x)g(x)]=LM\lim_{x\to a}[f(x)g(x)]=LM
Limit product lawThe limit of f of x as x approaches a equals L. The limit of g of x as x approaches a equals M. L and M are finite real numbers for the form shown.

Use this law to multiply two component limits that both exist.

lim⁡x→af(x)g(x)=LM\lim_{x\to a}\frac{f(x)}{g(x)}=\frac{L}{M}
Limit quotient lawThe limit of f of x as x approaches a equals L. The limit of g of x as x approaches a equals M. M is not zero.

Use this law to divide two component limits when the denominator limit is nonzero.

lim⁡x→0sin⁡xx=1\lim_{x\to 0}\frac{\sin x}{x}=1
Standard sine limitAngles are measured in radians. The limit is two-sided over the real numbers.

Use this standard limit when a trigonometric expression reduces to sine of a small angle divided by that angle.

Notation

Symbols used on this sheet

SymbolMeaningUsage note
f,gf,gFunctionsAssume the differentiability or continuity stated beside the formula.
f′f'Derivative of fThe prime uses the variable named in the surrounding expression.
CCConstant of integrationRequired for an indefinite integral family.
a,ba,bInterval endpointsOrientation matters: reversing them changes the sign of a definite integral.
nnConstant exponentSome power formulas exclude particular values such as n equals minus one.
lim⁡x→a\lim_{x\to a}Limit as x approaches aA plus or minus superscript identifies a one-sided approach.

Use the reference

Short applications

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

Choose the product rule

The expression multiplies a polynomial by an exponential, so differentiating both factors separately is not enough.

ddx(x2ex)\frac{d}{dx}(x^2e^x)
ex(x2+2x)e^x(x^2+2x)

Condition check: Expanding first is not possible, but a numerical difference quotient agrees at regular points.

Use the evaluation theorem

A continuous integrand has the antiderivative minus cosine x, which can be evaluated at both endpoints.

∫0πsin⁡x dx\int_0^{\pi}\sin x\,dx
[−cos⁡x]0π=2[-\cos x]_0^{\pi}=2

Condition check: The sine curve is nonnegative on the interval, so a positive accumulated value is consistent with its graph.

Use conditions, not memory alone

Common formula confusions

Using the wrong structure rule

Applying the product rule to a composition or the chain rule to a simple product.

Fix: Identify whether functions are added, multiplied, divided, or nested before choosing a rule.

Forgetting a condition

Using the power antiderivative with exponent minus one or cancelling a denominator at a forbidden point.

Fix: Read the condition beside the formula and carry the original domain into the result.

Dropping plus C

Writing one representative antiderivative as if it were the entire indefinite-integral answer.

Fix: Add an arbitrary constant after every indefinite integral.

Confusing value and method

Treating a numerical definite integral as though it were an antiderivative function.

Fix: Decide whether the task asks for a family of functions or one accumulated value.

Explore this collection

Open a focused formula reference

Use the subject sheet for broad review or open the narrower reference that matches the formula you need.

  • Formula reference

    Derivative formulas

    Match the visible function structure to a derivative rule, then confirm its domain and differentiability conditions.

  • Formula reference

    Integration formulas

    Find a matching antiderivative or theorem, preserve every condition, and verify the result by differentiation or endpoint evaluation.

  • Formula reference

    Limit formulas and laws

    Combine known limits only when their hypotheses hold, and use standard forms with the correct approach and angle measure.

Common questions

Calculus formulas FAQ

What is included in the calculus formula sheet?

The first complete edition covers core derivative rules, common antiderivatives, definite-integral evaluation, and standard limit relationships. Each category states the conditions needed to use its formulas.

Is the PDF different from the web page?

No separate formula copy is maintained. The HTML reference and PDF are generated from the same structured formula catalog so the notation, names, and conditions can be tested together.

Does a formula sheet replace learning the method?

No. A formula helps after you have identified the mathematical structure. Use the linked guide or calculator when you need to understand why a rule applies or see a complete example.

Why do some formulas need domain conditions?

Differentiability, continuity, a nonzero denominator, an exponent restriction, or radian angle measure can change whether a familiar-looking formula is valid.

Sources and curriculum alignment

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

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