I am using D to get derivatives of a function. However, R does not simplify the expression when returning the derivative. I need to figure out if a function has a derivative that can be expressed generically. Is there some way in R to simplify the expression?
> D(expression(sqrt(1 - x^2)), 'x')
-(0.5 * (2 * x * (1 - x^2)^-0.5))
> D(D(expression(sqrt(1 - x^2)), 'x'), 'x')
-(0.5 * (2 * (1 - x^2)^-0.5 - 2 * x * (-0.5 * (2 * x * (1 - x^2)^-1.5))))
Secondly, is there a way in R to do numerical integration?
Answer
library(Ryacas)
x <- Sym("x")
Simplify(deriv(sqrt(1 - x^2),x,2)) # return the result simplified
gives
expression((x^2 - 1 - x^2)/root(1 - x^2, 2)^3)
You can also try
PrettyForm(Simplify(deriv(sqrt(1 - x^2),x,2)))
which gives
2 2
x - 1 - x
---------------
3
/ 2 \
Sqrt\ 1 - x /
As for numerical integration try giving this to see what is available
library(sos)
findFn('{numerical+integration}')