Calculus

Calculus is the mathematical study of continuous change.

It has two main branches: differential calculus and integral calculus. These two branches are related by the fundamental theorem of calculus:

$\int_a^b f(x) \,\mathrm{d}x = F(b) - F(a)$

where $F$ is an antiderivative of $f$ , that is $F' = f$ .

To calculate the derivative of a function, use the D function or ND to calculate a numerical approximation

To calculate the integral (antiderivative) of a function, use the Integrate function or NIntegrate to calculate a numerical approximation.

["D", expr, var]

The D function represents the partial derivative of a function expr with respect to the variable var.

f^\prime(x)

$$$f^\prime(x)$$

["D", "f", "x"]

["D", expr, var-1, var-2, ...]

Multiple variables can be specified to compute the partial derivative of a multivariate function.

f^\prime(x, y)

$$$f^\prime(x, y)$$

f'(x, y)

$$$f'(x, y)$$

["D", "f", "x", "y"]

["D", expr, var, var]

A variable can be repeated to compute the second derivative of a function.

f^{\prime\prime}(x)

$$$f^{\prime\prime}(x)$$

f\doubleprime(x)

$$$f\doubleprime(x)$$

["D", "f", "x", "x"]

Explanation

The derivative is a measure of how a function changes as its input changes. It is the ratio of the change in the value of a function to the change in its input value.

The derivative of a function $f(x)$ with respect to its input $x$ is denoted by $f'(x)$ or $\frac{df}{dx}$ . The derivative of a function $f(x)$ is defined as:

f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}

where:

$h$ is the change in the input variable.
$f(x + h) - f(x)$ is the change in the value of the function.
$\frac{f(x + h) - f(x)}{h}$ is the ratio of the change in the value of the function to the change in its input value.
$\lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$ is the limit of the ratio of the change in the value of the function to the change in its input value as $h$ approaches $0$ .
The limit is taken as $h$ approaches $0$ because the derivative is the instantaneous rate of change of the function at a point, and the change in the input value must be infinitesimally small to be instantaneous.
Wikipedia: Derivative
Wolfram Mathworld: Derivative

Reference

Wikipedia: Derivative
Wikipedia: Notation for Differentiation, Leibniz's Notation, Lagrange's Notation, Newton's Notation
Wolfram Mathworld: Derivative
NIST: Derivative

Lagrange Notation

LaTeX	MathJSON
`f'(x)`	`["Derivative", "f", "x"]`
`f\prime(x)`	`["Derivative", "f", "x"]`
`f^{\prime}(x)`	`["Derivative", "f", "x"]`
`f''(x)`	`["Derivative", "f", "x", 2]`
`f\prime\prime(x)`	`["Derivative", "f", "x", 2]`
`f^{\prime\prime}(x)`	`["Derivative", "f", "x", 2]`
`f\doubleprime(x)`	`["Derivative", "f", "x", 2]`
`f^{(n)}(x)`	`["Derivative", "f", "x", "n"]`

["ND", expr, value]

The ND function returns a numerical approximation of the partial derivative of a function expr at the point value.

\sin^{\prime}(x)|_{x=1}

$$$\sin^{\prime}(x)|_{x=1}$$

["ND", "Sin", 1]
// ➔ 0.5403023058681398

Note: ["ND", "Sin", 1] is equivalent to ["N", ["D", "Sin", 1]].

["Derivative", expr]

The Derivative function represents the derivative of a function expr.

f^\prime(x)

$$$f^\prime(x)$$

["Apply", ["Derivative", "f"], "x"]

// This is equivalent to:
[["Derivative", "f"], "x"]

["Derivative", expr, n]

When a n argument is present it represents the n-th derivative of a function expr.

f^{(n)}(x)

$$$f^{(n)}(x)$$

["Apply", ["Derivative", "f", "n"], "x"]

Derivative is an operator in the mathematical sense, that is, a function that takes a function as an argument and returns a function.

The Derivative function is used to represent the derivative of a function in a symbolic form. It is not used to calculate the derivative of a function. To calculate the derivative of a function, use the D function or ND to calculate a numerical approximation.

["Integrate", expr]

An indefinite integral, also known as an antiderivative, refers to the reverse process of differentiation.

\int \sin

$$$\int \sin$$

["Integrate", "Sin"]

["Integrate", expr, index]

\int \sin x \,\mathrm{d}x

$$$\int \sin x \,\mathrm{d}x$$

["Integrate", ["Sin", "x"], "x"]

Note

The LaTeX expression above include a LaTeX spacing command \, to add a small space between the function and the differential operator. The differential operator is wrapped with a \mathrm{} command so it can be displayed upright. Both of these typographical conventions are optional, but they make the expression easier to read. The expression \int \sin x dx $\int f(x) dx$ is equivalent.

["Integrate", expr, bounds]

A definite integral computes the net area between a function $f(x)$ and the x-axis over a specified interval $[a, b]$ . The "net area" accounts for areas below the x-axis subtracting from the total.

\int_{0}^{2} x^2 \,\mathrm{d}x

$$$\int_{0}^{2} x^2 \,\mathrm{d}x$$

["Integrate", ["Power", "x", 2], ["Tuple", "x", 0, 2]]

The notation for the definite integral of $f(x)$ from $a$ to $b$ is given by:

\int_{a}^{b} f(x) \mathrm{d}x = F(b) - F(a)

where:

$dx$ indicates the variable of integration.
$[a, b]$ are the bounds of integration, with $a$ being the lower bound and $b$ being the upper bound.
$F(x)$ is an antiderivative of $f(x)$ , meaning $F'(x) = f(x)$ .

Use NIntegrate to calculate a numerical approximation of the definite integral of a function.

["Integrate", expr, bounds, bounds]

A double integral computes the net volume between a function $f(x, y)$ and the xy-plane over a specified region $[a, b] \times [c, d]$ . The "net volume" accounts for volumes below the xy-plane subtracting from the total. The notation for the double integral of $f(x, y)$ from $a$ to $b$ and $c$ to $d$ is given by:

\int_{a}^{b} \int_{c}^{d} f(x, y) \,\mathrm{d}x \,\mathrm{d}y

\int_{0}^{2} \int_{0}^{3} x^2 \,\mathrm{d}x \,\mathrm{d}y

$$$\int_{0}^{2} \int_{0}^{3} x^2 \,\mathrm{d}x \,\mathrm{d}y$$

["Integrate", ["Power", "x", 2], ["Tuple", "x", 0, 3], ["Tuple", "y", 0, 2]]

Explanation

Given a function $f(x)$ , finding its indefinite integral, denoted as $\int f(x) \,\mathrm{d}x$ , involves finding a new function $F(x)$ such that $F'(x) = f(x)$ .

Mathematically, this is expressed as:

\int f(x) \,\mathrm{d}x = F(x) + C

where:

$\mathrm{d}x$ specifies the variable of integration.
$F(x)$ is the antiderivative or the original function.
$C$ is the constant of integration, accounting for the fact that there are many functions that can have the same derivative, differing only by a constant.

Reference

Wikipedia: Integral, Antiderivative, Integral Symbol
Wolfram Mathworld: Integral
NIST: Integral

["NIntegrate", expr, lower-bound, upper-bound]

Calculate the numerical approximation of the definite integral of a function $f(x)$ from $a$ to $b$ .

\int_{0}^{2} x^2 \,\mathrm{d}x

$$$\int_{0}^{2} x^2 \,\mathrm{d}x$$

["NIntegrate", ["Function", ["Power", "x", 2], "x"], 0, 2]

["Limit", fn, value]

Evaluate the expression fn as it approaches the value value.

\lim_{x \to 0} \frac{\sin(x)}{x} = 1

$$$\lim_{x \to 0} \frac{\sin(x)}{x} = 1$$

["Limit", ["Divide", ["Sin", "_"], "_"], 0]

["Limit", ["Function", ["Divide", ["Sin", "x"], "x"], "x"], 0]

This function evaluates to a numerical approximation when using expr.N(). To get a numerical evaluation with expr.evaluate(), use NLimit.

["NLimit", fn, value]

Evaluate the expression fn as it approaches the value value.

["NLimit", ["Divide", ["Sin", "_"], "_"], 0]
// ➔ 1

["NLimit", ["Function", ["Divide", ["Sin", "x"], "x"], "x"], 0]
// ➔ 1

The numerical approximation is computed using a Richardson extrapolation algorithm.