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Nerdamer: Calculus Operations

JS Calc provides access to several symbolic calculus operations via Nerdamer.js.

Differentiation: diff()

Calculates the symbolic derivative of an expression with respect to a variable.

Complex definite integrals requiring advanced numerical methods might be blocked by CSP.

Syntax: diff(expression, variable [, order])

  • expression: The expression to differentiate.
  • variable: The variable with respect to which differentiation is performed.
  • order (Optional): A positive integer specifying the order of differentiation (e.g., 2 for the second derivative). Defaults to 1.

Examples:

> diff(x^3 - sin(x), x)
 3*x^2-cos(x)

> diff(2t^2 + 3t + 5, t) // diff(at^2 + bt + c, t) = 2at+b
 3+4*t

> diff(x^4, x, 2) // Second derivative
 12*x^2

> diff(exp(ky), y, 3) // Third derivative
  e^(k*y)*k^3

Indefinite Integration: integrate()

Calculates the symbolic indefinite integral (antiderivative) of an expression with respect to a variable.

Syntax: integrate(expression, variable)

  • Nerdamer typically does not add the constant of integration (+ C).

Examples:

> integrate(x^2 + 2x, x)
 (1/3)*x^3+x^2

> integrate(cos(y), y)
 sin(y)

> integrate(1/t, t)
 log(t)

Definite Integration: defint()

Calculates the symbolic definite integral of an expression.

Syntax: defint(expression, variable, lower_bound, upper_bound)

Examples:

> defint(x^2, x, 0, 2)
 8/3

> defint(exp(x), x, 0, 1)
 -1+e // or e-1
  • CSP Note: Very complex definite integrals, especially those involving non-elementary functions, might be blocked by browser Content Security Policy like defint(e^(cos(x)), x, 1, 2,).

Limits: limit()

Computes the limit of an expression as a variable approaches a certain point.

Syntax: limit(expression, variable, point)

Examples:

> limit(sin(x)/x, x, 0)
 1

> limit((x^2-1)/(x-1), x, 1)
 2

limit(1/x, x, infinity)
// Approaches 0

> limit(1/x, x, 1)
 1

> limit(1/x, x, 10)
 0.1

> limit(1/x, x, 100)
 0.01

> limit(1/x, x, 1000000)
 1e-6

limit((1+1/n)^n, n, infinity)
// Definition of and approaches e = 2.718281828459.

> limit((1+1/n)^n, n, 10)
 2.5937424601

> limit((1+1/n)^n, n, 100)
 2.7048138294215

Symbolic Summation: sum()

Computes the symbolic sum of an expression (Sigma notation).

Syntax: sum(expression, index_variable, lower_bound, upper_bound)

Examples:

sum(k, k, 1, n) // 1 + 2 + 3 + ... + n or (1/2)n(n+1)
> sum(k, k, 1, 253) // (1/2)*253*(253+1)
 32131

> sum(j^2, j, 1, 5) // 1^2 + 2^2 + 3^2 + 4^2 + 5^2
 55

> sum(1/2^i, i, 0, 3) // Geometric series
 15/8

Symbolic Product: product()

Computes the symbolic product of an expression (Pi notation).

Syntax: product(expression, index_variable, lower_bound, upper_bound)

Examples:

> product(k, k, 1, n) // n! (Factorial)
> factorial(n) // Or Nerdamer might expand for small n

> product(i, i, 1, 4) // 1*2*3*4
 24