What are you looking for when you ask for integrals and derivatives? Numerical or symbolic? Indefinite or definite? Single-variable or multivariate? Etc.
It's actually not hard to write a program to do symbolic differentiation, though simplification and display is far from simple (contrary, perhaps, to intuition). Similarly, numerical differentiation and numerical definite integration is simple enough, depending on how much precision you need. (If you need a little it's easy; if you need a fair bit and are doing many operations, you need some serious numerical analysis (math) to make it work; if you need lots of precision you need a bignum library so JS isn't that great.)
'Of course' symbolic integration is extremely difficult in any language. Your best bet in that case would be to query a site like Wolfram|alpha...
If you need clarification or expansion on any of these points, let me know.
