Find the first derivative of y = x ^{ x} for x > 0 with all the steps presented.

Note that the function y = x ^{ x} is neither a power function of the form x ^{ k} nor an exponential function of the form b ^{ x} and the known formulas of
Differentiation of these two functions
cannot be used. We need to find another method to find the first derivative of the given function.

Given

Take the natural log (ln) of both sides of the above

Use properties of logarithmic functions ln A^{b} = b ln A to the right side of the above equation and obtain

Differentiate both sides of the above with respect to x , using the chain rule on the left side and the product rule on the right.

Simplify the right side

Multiply both sides by y and simplify

Substitute y by x ^{ x} to obtain the final answer

Find the first derivative of

__Answer to the Above Exercise:__