Use Logarithmic Differentiation to Find the Derivative of the Function y = x^x

To find the derivative of the function y = x^x, we can use logarithmic differentiation. This technique is especially useful when dealing with functions where the variable appears in both the base and the exponent.

First, we take the natural logarithm of both sides:

ln(y) = ln(x^x)

Using the properties of logarithms, we can simplify the right side:

ln(y) = x * ln(x)

Next, we differentiate both sides with respect to x. Remember to apply the chain rule on the left side:

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

On the right side, we need to use the product rule:

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

Now, we substitute this back into our differentiation:

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

To solve for dy/dx, we multiply both sides by y:

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

Since we initially set y = x^x, we can substitute back:

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

Thus, the derivative of the function y = x^x is:

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