I've been challenging myself with finding tricky derivatives lately, and I'm stuck with this one. Does anyone have a good way to differentiate x^x? I tried the difference quotient and you used the concept of pascal's triangle to try to simplify terms such as (x+h)^(x+h) or (x+h)^x(x+h)^h and of course this got incredibly messy.
Another strange thing I'm finding is that when I use a derivative calculator online it shows: x^xlog(x)+x^x. However, when I use my ti-83 to test dy/dx at a given x value it's yielding a different value. I tried x=3, and according to the derivative calculator dy/dx should equal about 39.8. But my ti-83 shows 56.6 at x=3 when using dy/dx on the graph.
This isn't a homework problem. Sorry if I don't know how to approach the problem correctly.