1. Consider the function y = xx (for x > 0).

Question

a) Why does the derivative rule for xn not apply?
b) Why does the derivative rule for ax not apply?
c) What are the four magic properties of logarithms?
d) Take the ln of both sides. Use the properties of logarithms and implicit differentiation to determine y. Express your answer as a function of x only.

Answer 1

You must mean y = x^x or y = x^x

take ln of both sides
ln y = ln (x^x)
ln y = x lnx
now differentiate:

y'/y = x(1/x) + (1)lnx = 1 + lnx
y' = y(1 + lnx) = y + lnx
or
y' = x^x + lnx