clear all
; Some symbolic differentiation examples follow.

; Take the derivative of the absolute value function:
|x|
derivative ; The result is the sign function sgn(x), which gives the sign of x.
repeat echo *
; Mathomatic can differentiate anything that doesn't require symbolic logarithms.
y=e^(1+1/x)
derivative ; The first order derivative is:
derivative ; The second order derivative is:
expand fraction ; Perhaps easier to read:
repeat echo *
; A Taylor series demonstration:
y=x_new^n ; x_new is what we want, without using the root operator.
x_new ; It is easily solved for in Mathomatic.
y ; But we want an algorithm to compute it without using non-integer exponentiation.
taylor x_new, 1, x_old ; build the (nth root of y) iterative approximation formula
solve verifiable x_new ; solve for the output variable
; That is the convergent nth root approximation formula.
copy ; "calculate x_old 10000" tests this formula, if you would like to see for yourself.
replace x_old x_new with x ; make x_old (input) and x_new (output) the same
x ; make sure the formula was correct by solving for x
repeat echo *
; Another Taylor series demo:
e^x ; enter the exponential function
taylor x, 10, 0 ; generate a 10th order taylor series of the exponential function
laplace x ; do a Laplace transform on it
simplify ; show the structure of the result
laplace inverse x ; undo the Laplace transform
compare with 10 ; check the result