All knowing Massassians, how would I derive y = x^(x+1) ? Do I have to take log base x on both sides?

exponential rule

y' = x^x(ln(x)x + x + 1)

Take the natural log (ln) of both sides of the equation.

ln(y) = ln(x^(x+1)) = (x+1) ln(x)

Then do d/dx of both sides.

By the chain rule on the right:

d(ln(y))/dx = (1/y) * dy/dx

By the product rule on the left:

d((x+1)ln(x))/dx = ln(x) + (x+1)/x


Then putting the two sides back together:

(1/y)*dy/dx = ln(x) + (x+1)/x

or

dy/dx = y * (ln(x) + (x+1)/x)

But we have y explicitly defined in terms of x, so:

dy/dx = x^(x+1) * (ln(x) + (x+1)/x)