Calculus Operations

Differentiate()

The formats:
  • differentiate(expression)
  • differentiate(expression, var)
but the variable arument may be ommitted if there is only one variable in the expresstion or if variable x is intended. For example: differrentiate(sin(x)*exp(x)) partial differentials can also be performed by specifying the intended variable,in the form, differentiaite(function, variable), for example: differentiate(2x*y^2 + 3y + z,y) will yield a solution for when the fuction is differentiated with respect to y while other variables (x and z) are constants.

Integrate()

MathSend can provide solution to both definite and indefinite integrals to compute an integral. Formats:
  • integrate(expression)
  • integrate(expression, variable)
  • integrate(expression, variable, lower_limit, upper_limit)
For example, integrate(3*y*x^4+8y) will integrate expression with respect to x, while for other variables the form integrate(3*y/x^4+8yxz, y) will integrate with respect to y. Multiple integrals can also be performed by specifying the order of the variable, for example integrate integrate(3*y/x^4+8yxz, y, x,x,z integrates the expression four times: first with respect to y, second with respect to x, third with respect to x again, and fourth with respect to z. For definite integrals, the format is : integrate(expression, variable, lower_limit, upper_limit), for example integrate(3*y/x^4+8yxz, y, 0, inf) will integrate the expression with respect to y from 0 to infinity

Limit()

Mathsend utilize the Sympy module to compute Limits to a supplied fuction using Gruntz heuristics and algorithm. Formats:
  • limit(expression)
  • limit(expression, variable)
  • limit(expression, variable, lower_limit, upper_limit)
The format is limit(expression, variable, lower_limit, upper_limit). For example: limit(x*sin(1/x), x, inf)

Series()

Asymptotic series expansions can be computed with the series keywords. The formats:
  • series(expression)
  • series(expression, variable)
  • series(expression, variable, x0, upper_limit)
if the x0 ans order values are not specified (that is, any of the first two formats) x0 will be taken as 0 while the order as 6. For example: series(sin(x)) computes the Taylor series of sin(x) up to the order of series; and for a higher order of the series, say 9, use series(sin(x),0,9). For a different initial value, say 2, with the solution required up to the 9th order of x, we have series(sin(x), x, 2, 9).