In calculus, the derivative of a function is the slope of the tangent line at any point on the function. In this problem, we are asked to find the derivative of the function y = log(x^x). To find the derivative, we will use the chain rule, which is a method for finding the derivative of composite functions.



To use the chain rule, we need to break the function down into two parts: the outer function and the inner function. In this case, the outer function is y = log(u) and the inner function is u = x^x.

Next, we need to find the derivative of the inner function u with respect to x. To do this, we can use the exponential rule for derivatives:

u = x^x
ln(u) = x ln(x) (take the natural logarithm of both sides)
du/dx = u (ln(x) + 1) (take the derivative of both sides using the chain rule)

Now that we have the derivative of the inner function u, we can use the chain rule to find the derivative of the outer function y with respect to x:

dy/dx = dy/du * du/dx

To find dy/du, we can use the derivative of the natural logarithm function:

y = log(u)
dy/du = 1/u

Putting it all together, we have:

dy/dx = dy/du * du/dx
dy/dx = (1/u) * u(ln(x) + 1)
dy/dx = ln(x) + 1



The derivative of the function y = log(x^x) with respect to x is ln(x) + 1. This means that the slope of the tangent line at any point on the function is equal to ln(x) + 1.