Calculus topic hub
Learn limits, derivatives, integrals, sums, products, differentials, and common calculus formulas.
Topic overview
Read the operator, bounds, variable, differential, and domain together; then verify the result by differentiation, integration, or a limiting case.
Recommended order
Identify the operator and variable
Read bounds and indexes
Apply the matching rule
Verify with an inverse operation or known case
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Avoid these errors
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Step-by-step explanations
LaTeX
Build indexed limits, summations, and products with correct lower and upper bounds in inline and display math.
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Write one-sided, infinite, and multivariable limits with correct subscript notation.
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Add indices, limits, conditions, and aligned multi-line sums.
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Format definite, indefinite, multiple, contour, and line integrals.
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Insert integral signs in Word, Windows, macOS, HTML, Unicode, and LaTeX without losing limits, differentials, or mathematical structure.
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Create summation notation with limits in plain text, Word equations, Markdown, and LaTeX.
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Differentiate polynomial terms using the power rule, constants, sums, and a substitution check against a difference quotient.
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Find basic antiderivatives term by term, include the constant of integration, and verify by differentiating the result.
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\int\partial\infty\sum\langle\rangle\doteq\cdot⨝Variables and assumptions
Calculus
\int u\,dv=uv-\int v\,duTransforms an integral of a product using the product rule in reverse.
Open formulaCalculus
f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}Defines the instantaneous rate of change of a function as a limit.
Open formulaCalculus
\int_a^b f(x)\,dx=F(b)-F(a),\quad F'=fConnects definite integration with antiderivatives.
Open formulaCalculus
(fg)'=f'g+fg'Differentiates the product of two differentiable functions.
Open formulaCalculus
\frac{d}{dx}f(g(x))=f'(g(x))g'(x)Differentiates a composite function by combining outer and inner derivatives.
Open formulaAlgebra
S_n=\frac n2(a_1+a_n)Computes the sum of the first n terms of an arithmetic sequence.
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Identify the operator and variable
Read calculus notation Write calculus in LaTeX Use and verify common formulas
Omitting the differential