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A complete calculus study companion: browse 136 formulas with worked examples, drill yourself with tracked quizzes, review reference tables and constants, convert units, and evaluate expressions — all in your browser.

136

formulas across 8 calculus topics

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Limit Definition (Epsilon-Delta)

lim⁡x→af(x)=L  ⟺  ∀ϵ>0,∃δ>0:0<∣x−a∣<δ⇒∣f(x)−L∣<ϵ\lim_{x \to a} f(x) = L \iff \forall \epsilon > 0, \exists \delta > 0 : 0 < |x - a| < \delta \Rightarrow |f(x) - L| < \epsilon

The formal epsilon-delta definition of a limit. For every epsilon greater than zero, there exists a delta such that f(x) is within epsilon of L whenever x is within delta of a.

Limits & Continuity · Worked examples

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Left-Hand Limit

lim⁡x→a−f(x)=L\lim_{x \to a^-} f(x) = L

The limit of f(x) as x approaches a from the left (from values less than a).

Limits & Continuity · Worked examples

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Right-Hand Limit

lim⁡x→a+f(x)=L\lim_{x \to a^+} f(x) = L

The limit of f(x) as x approaches a from the right (from values greater than a). A two-sided limit exists if and only if both one-sided limits exist and are equal.

Limits & Continuity · Worked examples

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Squeeze Theorem

g(x)≤f(x)≤h(x) and lim⁡x→ag(x)=lim⁡x→ah(x)=L⇒lim⁡x→af(x)=Lg(x) \leq f(x) \leq h(x) \text{ and } \lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L \Rightarrow \lim_{x \to a} f(x) = L

If f(x) is squeezed between g(x) and h(x) near a, and g and h have the same limit L, then f also has limit L.

Limits & Continuity · Worked examples

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Limit of a Sum

lim⁡x→a[f(x)+g(x)]=lim⁡x→af(x)+lim⁡x→ag(x)\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)

The limit of a sum equals the sum of the limits, provided both limits exist.

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Limit of a Product

lim⁡x→a[f(x)⋅g(x)]=lim⁡x→af(x)⋅lim⁡x→ag(x)\lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)

The limit of a product equals the product of the limits, provided both limits exist.

Limits & Continuity · Worked examples

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Limit of a Quotient

lim⁡x→af(x)g(x)=lim⁡x→af(x)lim⁡x→ag(x)\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}

The limit of a quotient equals the quotient of the limits, provided the denominator limit is nonzero.

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Limit of a Power

lim⁡x→a[f(x)]n=[lim⁡x→af(x)]n\lim_{x \to a} [f(x)]^n = \left[\lim_{x \to a} f(x)\right]^n

The limit of a power equals the power of the limit, for any positive integer n.

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Limit of a Constant Multiple

lim⁡x→a[c⋅f(x)]=c⋅lim⁡x→af(x)\lim_{x \to a} [c \cdot f(x)] = c \cdot \lim_{x \to a} f(x)

A constant factor can be pulled out of a limit.

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Limit of sin(x)/x

lim⁡x→0sin⁡xx=1\lim_{x \to 0} \frac{\sin x}{x} = 1

One of the most important special limits in calculus. Often proved using the Squeeze Theorem with geometric arguments.

Limits & Continuity · Worked examples

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Limit of (cos x - 1)/x

lim⁡x→0cos⁡x−1x=0\lim_{x \to 0} \frac{\cos x - 1}{x} = 0

A special limit related to the derivative of cosine at x = 0.

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Limit Definition of e

lim⁡n→∞(1+1n)n=e\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e

The number e (≈ 2.71828) defined as a limit. Equivalently, lim(x→0) (1+x)^(1/x) = e.

Limits & Continuity · Worked examples

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L'Hopital's Rule

If 00 or ±∞±∞:lim⁡x→af(x)g(x)=lim⁡x→af′(x)g′(x)\text{If } \frac{0}{0} \text{ or } \frac{\pm\infty}{\pm\infty}: \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}

L'Hopital's Rule: When a limit gives an indeterminate form 0/0 or ∞/∞, the limit equals the ratio of the derivatives (if that limit exists).

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Continuity Definition

f is continuous at a  ⟺  lim⁡x→af(x)=f(a)f \text{ is continuous at } a \iff \lim_{x \to a} f(x) = f(a)

A function is continuous at a point a if (1) f(a) is defined, (2) lim(x→a) f(x) exists, and (3) the limit equals f(a).

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Intermediate Value Theorem

f continuous on [a,b],  f(a)<N<f(b)⇒∃ c∈(a,b):f(c)=Nf \text{ continuous on } [a,b],\; f(a) < N < f(b) \Rightarrow \exists\, c \in (a,b) : f(c) = N

If f is continuous on [a,b] and N is between f(a) and f(b), then there exists at least one c in (a,b) where f(c) = N. Often used to show a root exists.

Limits & Continuity · Worked examples

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Extreme Value Theorem

f continuous on [a,b]⇒f attains an absolute max and min on [a,b]f \text{ continuous on } [a,b] \Rightarrow f \text{ attains an absolute max and min on } [a,b]

If f is continuous on a closed interval [a, b], then f attains both an absolute maximum and an absolute minimum on that interval.

Limits & Continuity · Worked examples

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Limits at Infinity

lim⁡x→∞anxn+⋯bmxm+⋯={0n<man/bmn=m±∞n>m\lim_{x \to \infty} \frac{a_n x^n + \cdots}{b_m x^m + \cdots} = \begin{cases} 0 & n < m \\ a_n/b_m & n = m \\ \pm\infty & n > m \end{cases}

For rational functions as x→∞, compare the degrees of numerator (n) and denominator (m) to determine the limit.

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Infinite Limits (Vertical Asymptotes)

lim⁡x→af(x)=±∞⇒x=a is a vertical asymptote\lim_{x \to a} f(x) = \pm\infty \Rightarrow x = a \text{ is a vertical asymptote}

If f(x) approaches ±∞ as x approaches a, then the line x = a is a vertical asymptote of f.

Limits & Continuity · Worked examples

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Limit Definition of Derivative

f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

The derivative of f at x is defined as the limit of the difference quotient as h approaches 0.

Derivatives · Worked examples

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Constant Rule

ddx[c]=0\frac{d}{dx}[c] = 0

The derivative of any constant is zero.

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Power Rule

ddx[xn]=nxn−1\frac{d}{dx}[x^n] = nx^{n-1}

Bring the exponent down as a coefficient and reduce the exponent by one. Works for any real number n.

Derivatives · Worked examples

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Constant Multiple Rule

ddx[c⋅f(x)]=c⋅f′(x)\frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x)

A constant factor passes through the derivative operator.

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Sum/Difference Rule

ddx[f(x)±g(x)]=f′(x)±g′(x)\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)

The derivative of a sum or difference is the sum or difference of the derivatives.

Derivatives · Worked examples

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Product Rule

ddx[f(x)⋅g(x)]=f′(x)g(x)+f(x)g′(x)\frac{d}{dx}[f(x) \cdot g(x)] = f'(x)g(x) + f(x)g'(x)

The derivative of a product: derivative of the first times the second, plus the first times the derivative of the second.

Derivatives · Worked examples

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Quotient Rule

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}

The derivative of a quotient: (derivative of top times bottom minus top times derivative of bottom) over bottom squared.

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Chain Rule

ddx[f(g(x))]=f′(g(x))⋅g′(x)\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)

For composite functions: differentiate the outer function evaluated at the inner function, then multiply by the derivative of the inner function.

Derivatives · Worked examples

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Derivative of sin(x)

ddx[sin⁡x]=cos⁡x\frac{d}{dx}[\sin x] = \cos x

The derivative of the sine function is the cosine function.

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Derivative of cos(x)

ddx[cos⁡x]=−sin⁡x\frac{d}{dx}[\cos x] = -\sin x

The derivative of cosine is negative sine.

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Derivative of tan(x)

ddx[tan⁡x]=sec⁡2x\frac{d}{dx}[\tan x] = \sec^2 x

The derivative of tangent is secant squared.

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Derivative of cot(x)

ddx[cot⁡x]=−csc⁡2x\frac{d}{dx}[\cot x] = -\csc^2 x

The derivative of cotangent is negative cosecant squared.

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Derivative of sec(x)

ddx[sec⁡x]=sec⁡xtan⁡x\frac{d}{dx}[\sec x] = \sec x \tan x

The derivative of secant is secant times tangent.

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Derivative of csc(x)

ddx[csc⁡x]=−csc⁡xcot⁡x\frac{d}{dx}[\csc x] = -\csc x \cot x

The derivative of cosecant is negative cosecant times cotangent.

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Derivative of eˣ

ddx[ex]=ex\frac{d}{dx}[e^x] = e^x

The exponential function eˣ is its own derivative -a unique property of the natural exponential.

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Derivative of aˣ

ddx[ax]=axln⁡a\frac{d}{dx}[a^x] = a^x \ln a

The derivative of a general exponential function aˣ is aˣ times the natural log of the base.

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Derivative of ln(x)

ddx[ln⁡x]=1x\frac{d}{dx}[\ln x] = \frac{1}{x}

The derivative of the natural logarithm is 1/x.

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Derivative of log_a(x)

ddx[log⁡ax]=1xln⁡a\frac{d}{dx}[\log_a x] = \frac{1}{x \ln a}

The derivative of a logarithm with base a. Use the change of base formula: log_a(x) = ln(x)/ln(a).

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Derivative of arcsin(x)

ddx[arcsin⁡x]=11−x2\frac{d}{dx}[\arcsin x] = \frac{1}{\sqrt{1 - x^2}}

The derivative of the inverse sine function.

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Derivative of arccos(x)

ddx[arccos⁡x]=−11−x2\frac{d}{dx}[\arccos x] = -\frac{1}{\sqrt{1 - x^2}}

The derivative of the inverse cosine function. Note the negative sign compared to arcsin.

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Derivative of arctan(x)

ddx[arctan⁡x]=11+x2\frac{d}{dx}[\arctan x] = \frac{1}{1 + x^2}

The derivative of the inverse tangent function. Valid for all real x.

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Derivative of arccot(x)

ddx[arccot x]=−11+x2\frac{d}{dx}[\text{arccot}\, x] = -\frac{1}{1 + x^2}

The derivative of the inverse cotangent function. Note the negative sign compared to arctan.

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Derivative of arcsec(x)

ddx[arcsec x]=1∣x∣x2−1\frac{d}{dx}[\text{arcsec}\, x] = \frac{1}{|x|\sqrt{x^2 - 1}}

The derivative of the inverse secant function.

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Derivative of arccsc(x)

ddx[arccsc x]=−1∣x∣x2−1\frac{d}{dx}[\text{arccsc}\, x] = -\frac{1}{|x|\sqrt{x^2 - 1}}

The derivative of the inverse cosecant function. Note the negative sign, mirroring the arcsec derivative.

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Tangent Line

y−f(a)=f′(a)(x−a)y - f(a) = f'(a)(x - a)

The equation of the tangent line to f(x) at the point (a, f(a)). The slope is the derivative evaluated at x = a.

Applications of Derivatives · Worked examples

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Normal Line

y−f(a)=−1f′(a)(x−a)y - f(a) = -\frac{1}{f'(a)}(x - a)

The normal line is perpendicular to the tangent line. Its slope is the negative reciprocal of the derivative.

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Critical Points

f′(c)=0 or f′(c) DNE⇒c is a critical pointf'(c) = 0 \text{ or } f'(c) \text{ DNE} \Rightarrow c \text{ is a critical point}

Critical points occur where the derivative is zero or undefined. These are candidates for local extrema.

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First Derivative Test

f′ changes +→−⇒local max;f′ changes −→+⇒local minf' \text{ changes } + \to - \Rightarrow \text{local max}; \quad f' \text{ changes } - \to + \Rightarrow \text{local min}

At a critical point c: if f' changes from positive to negative, c is a local max. If f' changes from negative to positive, c is a local min.

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Second Derivative Test

f′(c)=0:f′′(c)>0⇒local min;f′′(c)<0⇒local maxf'(c) = 0: \quad f''(c) > 0 \Rightarrow \text{local min}; \quad f''(c) < 0 \Rightarrow \text{local max}

At a critical point where f'(c) = 0: if f''(c) > 0, c is a local minimum (concave up). If f''(c) < 0, c is a local maximum (concave down). If f''(c) = 0, the test is inconclusive.

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Concavity

f′′(x)>0⇒concave up;f′′(x)<0⇒concave downf''(x) > 0 \Rightarrow \text{concave up}; \quad f''(x) < 0 \Rightarrow \text{concave down}

The second derivative determines concavity. Concave up means the curve opens upward (bowl shape). Concave down means it opens downward.

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Inflection Points

f′′(c)=0 (or DNE) and concavity changes at c⇒(c,f(c)) is an inflection pointf''(c) = 0 \text{ (or DNE) and concavity changes at } c \Rightarrow (c, f(c)) \text{ is an inflection point}

An inflection point is where the concavity changes. Find candidates where f''(x) = 0 or is undefined, then verify concavity changes.

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Mean Value Theorem

f′(c)=f(b)−f(a)b−a for some c∈(a,b)f'(c) = \frac{f(b) - f(a)}{b - a} \text{ for some } c \in (a, b)

If f is continuous on [a,b] and differentiable on (a,b), then there exists at least one c in (a,b) where the instantaneous rate of change equals the average rate of change.

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Rolle's Theorem

f(a)=f(b)⇒∃ c∈(a,b):f′(c)=0f(a) = f(b) \Rightarrow \exists\, c \in (a,b) : f'(c) = 0

If f is continuous on [a,b], differentiable on (a,b), and f(a) = f(b), then there exists at least one c in (a,b) where f'(c) = 0. This is a special case of MVT.

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Related Rates

ddt[equation] -differentiate both sides with respect to time\frac{d}{dt}[\text{equation}] \text{ -differentiate both sides with respect to time}

Related rates problems involve finding how fast one quantity changes given how fast another changes. Differentiate an equation relating the quantities with respect to time.

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Optimization

Find critical points of f′(x)=0, then verify max/min via endpoint or derivative test\text{Find critical points of } f'(x) = 0 \text{, then verify max/min via endpoint or derivative test}

Optimization finds the maximum or minimum value of a function. Set up the objective function, find critical points, and test them.

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Linearization

L(x)=f(a)+f′(a)(x−a)L(x) = f(a) + f'(a)(x - a)

The linearization of f at a is the tangent line, used as a linear approximation of f near a. This is the same as the first-degree Taylor polynomial.

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Differentials

dy=f′(x) dxdy = f'(x)\, dx

The differential dy approximates the change in y for a small change dx. Used for error estimation and approximation.

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Newton's Method

xn+1=xn−f(xn)f′(xn)x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}

Newton's method iteratively approximates roots of f(x) = 0. Each step uses the tangent line to get a better approximation.

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L'Hopital's Rule (Applications)

lim⁡x→af(x)g(x)=00,∞∞lim⁡x→af′(x)g′(x)\lim_{x \to a} \frac{f(x)}{g(x)} \overset{\frac{0}{0}, \frac{\infty}{\infty}}{=} \lim_{x \to a} \frac{f'(x)}{g'(x)}

L'Hopital's Rule applied to evaluate limits involving indeterminate forms such as 0·∞, ∞-∞, 0⁰, ∞⁰, 1^∞ by algebraic rearrangement to 0/0 or ∞/∞.

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Riemann Sum

∫abf(x) dx=lim⁡n→∞∑i=1nf(xi∗)Δx\int_a^b f(x)\, dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x

The definite integral is defined as the limit of Riemann sums. Partition [a,b] into n subintervals of width Δx = (b-a)/n.

Integrals · Worked examples

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FTC Part 1

ddx[∫axf(t) dt]=f(x)\frac{d}{dx}\left[\int_a^x f(t)\, dt\right] = f(x)

The Fundamental Theorem of Calculus Part 1: The derivative of the integral (with variable upper limit) of a continuous function is the original function.

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FTC Part 2

∫abf(x) dx=F(b)−F(a) where F′(x)=f(x)\int_a^b f(x)\, dx = F(b) - F(a) \text{ where } F'(x) = f(x)

The Fundamental Theorem of Calculus Part 2: A definite integral can be evaluated using any antiderivative F of f.

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