calculus.js

Function.prototype.diff = function(eps) {
    var fn = this, eps = eps || 1e-6;
    return function(x) {
        return (fn(x+eps) - fn(x)) / eps;
    };
};

(function(x) { return x*x }).diff()(6)
// -> 12

Function.prototype.int = function(n) {
    var fn = this, n = n || 10000;
    return function(a,b) {
        var s = 0, i, eps = (b-a)/n;
        for (i = 0; i < n; i ++)
            s += fn(a + i*eps) * eps;
        return s;
    };
};

(function(x) { return x*x }).int()(0,4)
// -> 21.33