Differentiating the common expressions is quite easy but differentiation of terms having powers ie x^x involves logarithmic concepts.
Heres one of my observation in doing so:

If u have a general term, say y= a^b
Differentiating with respect to x we get:
dy/dx= a^b {b/a. da/dx + log(a).db/dx}

Couldn't make it more simple due to malfunctioning of formula editor.
Hope yoll find it useful.
Pls comment.