Calculus — Math expr, Chain rule (extended) Cheat Sheet

The core ideas of Calculus — Math expr, Chain rule (extended) distilled into a single, scannable reference — perfect for review or quick lookup.

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Quick Reference

Limits

The value that a function approaches as the input approaches some value. Limits are the foundation of calculus, used to define both derivatives and integrals.

Derivatives

The derivative measures the instantaneous rate of change of a function. Geometrically, it gives the slope of the tangent line to a curve at any point.

Integrals

Integration is the reverse process of differentiation. Definite integrals compute the signed area under a curve, while indefinite integrals find antiderivatives.

Fundamental Theorem of Calculus

Links differentiation and integration as inverse processes. Part 1: integration can be reversed by differentiation. Part 2: definite integrals can be computed using antiderivatives.

Chain Rule

A formula for computing the derivative of a composition of functions. If $y = f(g(x))$, then $\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$.

Power Rule

The derivative of $x^n$ is $nx^{n-1}$. This is the most frequently used differentiation rule.

Integration by Parts

A technique for evaluating integrals of products. Based on the product rule in reverse: $\int u\,dv = uv - \int v\,du$.

Taylor Series

An infinite sum of terms calculated from the derivatives of a function at a single point. Allows approximation of complex functions with polynomials.

Key Terms at a Glance

Antiderivative:A function $F$ whose derivative equals $f$. Also called an indefinite integral.

Chain Rule:A rule for differentiating composite functions: $\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$.

Continuity:A function is continuous at a point if its limit equals its value at that point.

Convergence:A sequence or series approaches a finite limit as more terms are added.

Definite Integral:The signed area under a curve between two bounds: $\int_a^b f(x)\,dx$.

Derivative:The instantaneous rate of change of a function, representing the slope of the tangent line.

Differential Equation:An equation involving derivatives of an unknown function.

Divergence:A sequence or series that does not approach a finite limit.

Extrema:Maximum or minimum values of a function, either local or absolute.

Fundamental Theorem of Calculus:The theorem linking differentiation and integration as inverse processes.

Indefinite Integral:The general antiderivative of a function, written with $+ C$.

Inflection Point:A point where the concavity of a function changes ($f''$ changes sign).

Integration:The process of finding the integral (antiderivative or area under curve) of a function.

L'Hopital's Rule:A method for evaluating limits of indeterminate forms ($\frac{0}{0}$ or $\frac{\infty}{\infty}$) using derivatives.

Limit:The value a function approaches as its input approaches a given value.

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