Calculus Formulas: Limits, Derivatives & Integrals

What Are The Calculus Formulas?

Calculus formulas are the standard rules for three linked operations: taking a limit, taking a derivative (a rate of change, read geometrically as the slope of a tangent line), and taking an integral (an accumulation, read geometrically as the area under a curve). A fourth group, the theorems, states exactly how derivatives and integrals connect. Nearly every problem in a first calculus course is an application of a formula from one of these four groups.

The groups build on each other in order:

• Limits define what "instantaneous" and "infinite sum" even mean. Every derivative and integral is secretly a limit.

• Derivatives measure how fast a quantity changes. The derivative of $f$ at a point is the slope of the tangent line there.

• Integrals measure how much accumulates. A definite integral is the signed area between a curve and the axis.

• Theorems (the Fundamental Theorem of Calculus and the Mean Value Theorem) tie the slope world and the area world into one subject.

The rest of this page states each group as a table you can use as a reference, pairs the algebra with its geometric meaning, and works a few small problems so the formulas do not sit as a bare list. For the wider map of the field, see the what is calculus.

What Are The Standard Limit Formulas?

A limit describes the value a function heads toward as its input approaches some point, written $\lim_{x \to a} f(x)$. These standard limits are the ones worth knowing by heart, because the derivative rules below are built from them.

Table: Standard limits used across calculus.

The first limit is the classic. As the angle shrinks, the arc $x$ and the vertical height $\sin x$ become the same length, so their ratio squeezes to $1$. For the full derivations and the limit laws that combine these, see limit formulas and standard limits.

What Are The Derivative Formulas?

The derivative $\frac{d}{dx}f(x)$ gives the slope of the tangent to $y = f(x)$ at each point, that is, the instantaneous rate of change. The table below is the core reference; a single consistent notation, the Leibniz operator $\frac{d}{dx}$, is used throughout.

Table: Derivatives of the standard functions.

Three rules combine these building blocks into the derivative of anything you can write down.

Table: The three combination rules for derivatives.

For the full derivations and every case, see the table of derivatives. Each combination rule has its own page: the power rule, the product rule, the quotient rule, and the chain rule. The general blueprint for all of them lives at the derivative reference.

What Are The Integration Formulas?

Integration reverses differentiation. An indefinite integral $\int f(x),dx$ is the family of antiderivatives of $f$, and it always carries a constant of integration $+C$. A definite integral $\int_a^b f(x),dx$ gives the signed area between the curve and the $x$-axis from $a$ to $b$.

Table: Standard indefinite integrals. Every one carries $+C$.

Read the power-rule integral against the power-rule derivative and you see they undo each other: differentiating $\frac{x^{n+1}}{n+1}$ returns $x^n$. The one row that breaks the pattern is $n = -1$, where the formula would divide by zero, so $\int x^{-1},dx = \ln\lvert x\rvert + C$ instead. For the full set and the methods that extend it, see the basic integration formulas, the table of integrals, and the general integration reference.

What Are The Main Theorems Of Calculus?

Two theorems turn the tables above from separate lists into one connected subject. Both have precise hypotheses that are easy to skip and costly to lose.

The Fundamental Theorem of Calculus. If $f$ is continuous on $[a, b]$ and $F$ is any antiderivative of $f$ (so $F' = f$), then

$$\int_a^b f(x),dx = F(b) - F(a).$$

This is why the integral tables matter: to find an area, find an antiderivative and subtract its values at the two ends. The theorem also runs the other way, stating that $\frac{d}{dx}\int_a^x f(t),dt = f(x)$, so differentiation and integration are inverse operations. The full treatment is at the Fundamental Theorem of Calculus.

The Mean Value Theorem. If $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$, then there is at least one point $c$ in $(a, b)$ where

$$f'(c) = \frac{f(b) - f(a)}{b - a}.$$

Geometrically, somewhere on a smooth curve the tangent line runs exactly parallel to the straight line joining the endpoints. On a drive that averages 60 km/h, at some instant your speedometer read exactly 60. The full statement and proof sit at the Mean Value Theorem.

What Are The Taylor And Maclaurin Series Formulas?

A Taylor series rebuilds a smooth function as an infinite polynomial around a centre $a$. Writing $n$ factorial for the product $1 \cdot 2 \cdot 3 \cdots n$, the series is

$$f(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2}(x - a)^2 + \frac{f'''(a)}{6}(x - a)^3 + \cdots$$

where the denominators $2, 6, 24, \ldots$ are $2$ factorial, $3$ factorial, $4$ factorial, and so on. Centre it at $a = 0$ and it is called a Maclaurin series. Three are worth knowing, each converging for every real $x$:

$$e^x = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} + \cdots$$

$$\sin x = x - \frac{x^3}{6} + \frac{x^5}{120} - \cdots$$

$$\cos x = 1 - \frac{x^2}{2} + \frac{x^4}{24} - \cdots$$

Each of these has an infinite radius of convergence, meaning the series equals the function for all $x$. For the general term, the convergence tests, and the remainder bound, see the Taylor series and Maclaurin series pages.

How Do You Use The Calculus Formulas? Worked Examples

Each example is fully stepped, and every integral is checked by differentiating the answer back to the integrand.

Example 1: Differentiate a product.

Find $\dfrac{d}{dx}\big(x^2 \sin x\big)$.

This is a product of $u = x^2$ and $v = \sin x$, so use the product rule with $u' = 2x$ and $v' = \cos x$:

$$\frac{d}{dx}\big(x^2 \sin x\big) = (2x)(\sin x) + (x^2)(\cos x) = 2x\sin x + x^2\cos x$$

Final answer: $2x\sin x + x^2\cos x$.

Example 2: Evaluate an indefinite integral.

Find $\displaystyle\int \big(3x^2 + 2\cos x\big),dx$.

Integrate term by term using the power rule and the cosine integral, then add one constant:

$$\int \big(3x^2 + 2\cos x\big),dx = x^3 + 2\sin x + C$$

Check: $\dfrac{d}{dx}\big(x^3 + 2\sin x + C\big) = 3x^2 + 2\cos x$, the original integrand, so the antiderivative is correct.

Final answer: $x^3 + 2\sin x + C$.

Example 3: Find an area with the Fundamental Theorem.

Find the area under $y = 3x^2$ from $x = 0$ to $x = 2$, that is $\displaystyle\int_0^2 3x^2,dx$.

An antiderivative of $3x^2$ is $F(x) = x^3$. Apply the Fundamental Theorem:

$$\int_0^2 3x^2,dx = \big[x^3\big]_0^2 = 2^3 - 0^3 = 8$$

Check: $\dfrac{d}{dx}\big(x^3\big) = 3x^2$, so $F$ is a valid antiderivative and the area is exact.

Final answer: the area is $8$ square units.

Example 4: Estimate a value with a Maclaurin series.

Estimate $e^{0.1}$ using the first three terms of the series for $e^x$.

Take $x = 0.1$ in $e^x \approx 1 + x + \frac{x^2}{2}$:

$$e^{0.1} \approx 1 + 0.1 + \frac{(0.1)^2}{2} = 1 + 0.1 + 0.005 = 1.1050$$

The true value is $e^{0.1} = 1.1052$ to four decimal places, so three terms already land within $0.0002$.

Final answer: $e^{0.1} \approx 1.1050$ (true value $1.1052$).

Why Do The Calculus Formulas Work?

The formulas are not arbitrary. Each one is a limit made concrete, and each carries a picture.

• A derivative is a slope. The derivative is the limit of the slope of a chord as its two points slide together, $\frac{d}{dx}f(x) = \lim_{h \to 0}\frac{f(x+h) - f(x)}{h}$. Feed the power function $x^n$ into that limit and the algebra collapses to $nx^{n-1}$, which is why the power rule looks the way it does.

• An integral is an area. The definite integral is the limit of a sum of thin rectangles under the curve. As the rectangles get thinner, the sum converges to the exact area, and the $+C$ on an indefinite integral records that area is only pinned down once you fix a starting point.

• The theorems close the loop. Because area grows at a rate equal to the height of the curve, differentiating an accumulated area returns the original function. That single fact, the Fundamental Theorem, is why the derivative table and the integral table are mirror images of each other.

Reading each formula as "a limit with a geometric meaning" is the difference between memorising the tables and understanding them. The limits and derivatives page develops this link step by step.

Who Discovered The Calculus Formulas?

The rules on this page were assembled over centuries, then sharpened into their modern form in the 1600s by two rivals working independently.

The formulas only became fully rigorous later, once limits had a precise definition:

• Augustin-Louis Cauchy (1789–1857, France) rebuilt calculus on the limit rather than on vague "infinitely small" quantities, giving the derivative and integral their careful definitions.

• Karl Weierstrass (1815–1897, Germany) supplied the exact epsilon-delta language for a limit, so the standard-limit table at the top of this page finally rested on solid ground.

The longer arc, from Archimedes to the modern textbook, is told at the history of calculus.

Where Are The Calculus Formulas Used In The Real World?

The same four groups of formulas run under a wide range of work.

• Physics and motion: velocity is the derivative of position and distance is the integral of velocity, so every kinematics problem is a derivative and an integral in disguise. See calculus in physics.

• Engineering: stress, heat flow, and signal analysis are all rates and accumulations, evaluated with the derivative and integral tables. See calculus in engineering.

• Economics: marginal cost and marginal revenue are derivatives, and total cost is recovered by integrating them. See calculus in economics.

• Computer graphics and machine learning: training a model minimises error by following derivatives (gradients), one of the largest uses of the chain rule ever built.

• Probability: the chance a continuous quantity falls in a range is the integral of its density function.

One compact toolkit of formulas describes motion, money, machines, and randomness, which is why calculus is a shared language across fields that otherwise share nothing.

What Are The Most Common Mistakes With Calculus Formulas?

These three errors account for most lost marks when students first use the formulas, and each matches a question real learners ask on r/calculus, r/learnmath, and course error handouts.

Using the power rule on an exponential function.

Where it slips in:

A student differentiates $e^x$ or $2^x$ as if it were a power, writing $\frac{d}{dx}e^x = xe^{x-1}$.

Don't do this:

Do not apply the power rule $nx^{n-1}$ to a variable exponent. The power rule is only for a variable base with a fixed power, like $x^n$.

The correct way:

Use the exponential rule. $\frac{d}{dx}e^x = e^x$, and for a general base $\frac{d}{dx}a^x = a^x\ln a$. The variable is in the exponent, so the base stays put.

Forgetting the constant of integration.

Where it slips in:

A student computes an indefinite integral such as $\int 2x,dx$ and writes $x^2$, stopping one symbol short.

Don't do this:

Do not drop the $+C$ on an indefinite integral. Infinitely many functions share the same derivative, and the constant records that.

The correct way:

Always write $\int 2x,dx = x^2 + C$. The $+C$ is only omitted for a definite integral, where it cancels in $F(b) - F(a)$.

Dropping the inner derivative in the chain rule.

Where it slips in:

A student differentiates $\sin(3x)$ as $\cos(3x)$ and stops, treating the inside as if it were a bare $x$.

Don't do this:

Do not ignore the derivative of the inner function. A function inside another function needs the chain rule.

The correct way:

Differentiate the outer function, then multiply by the derivative of the inside: $\frac{d}{dx}\sin(3x) = \cos(3x)\cdot 3 = 3\cos(3x)$.

Practice Problems On Calculus Formulas

Work each one, then check against the answer. Answers are verified.

1. Differentiate $f(x) = x^5$.
   (Answer: $\frac{d}{dx}x^5 = 5x^4$ by the power rule.)

2. Differentiate $g(x) = e^x\cos x$.
   (Answer: product rule gives $e^x\cos x - e^x\sin x = e^x(\cos x - \sin x)$.)

3. Evaluate $\displaystyle\int \big(4x^3 + \sec^2 x\big),dx$.
   (Answer: $x^4 + \tan x + C$; differentiate back to check.)

4. Evaluate $\displaystyle\int_1^2 6x,dx$.
   (Answer: $\big[3x^2\big]_1^2 = 12 - 3 = 9$.)

5. Find $\lim_{x \to 0}\dfrac{\sin 5x}{x}$.
   (Answer: rewrite as $5\cdot\frac{\sin 5x}{5x} \to 5\cdot 1 = 5$.)

6. Differentiate $h(x) = \tan^{-1} x$.
   (Answer: $\dfrac{1}{1 + x^2}$.)

Where Should You Go Next After Calculus Formulas?

This page is the hub of the calculus cluster, and several natural doors open from it.

1. Table of derivatives. The full derivative reference, with every standard function and each combination rule worked out.

2. Table of integrals. The companion integral reference, including the trigonometric and exponential cases this page only samples.

3. Fundamental Theorem of Calculus. The theorem that ties the two tables together, with both parts and their proofs.

If your child is building these foundations, a live Bhanzu trainer teaches the calculus formulas from the slope-and-area picture up, so the tables feel like one connected idea, in the Bhanzu math program.