Question

Differentiate: \( y = x^{x} \).

Hint:

For differentiating expressions of the form \( y = x^x \), use logarithmic differentiation.

Answer 1

To differentiate \( y = x^x \), we will use logarithmic differentiation.

Step 1: Take the natural logarithm of both sides.
\[ \ln y = \ln(x^x). \]

Step 2: Use the logarithmic identity \( \ln(a^b) = b \ln a \).
\[ \ln y = x \ln x. \]

Step 3: Differentiate both sides with respect to \( x \).
Using the product rule on the right-hand side: \[ \frac{d}{dx} \left( \ln y \right) = \frac{d}{dx} \left( x \ln x \right). \] On the left-hand side: \[ \frac{d}{dx} \left( \ln y \right) = \frac{1}{y} \frac{dy}{dx}. \] On the right-hand side, applying the product rule: \[ \frac{d}{dx} \left( x \ln x \right) = \ln x + 1. \] So, we have: \[ \frac{1}{y} \frac{dy}{dx} = \ln x + 1. \]

Step 4: Solve for \( \frac{dy}{dx} \).
Multiply both sides by \( y \): \[ \frac{dy}{dx} = y (\ln x + 1). \] Since \( y = x^x \), substitute back: \[ \frac{dy}{dx} = x^x (\ln x + 1). \] Conclusion:
The derivative of \( y = x^x \) is: \[ \boxed{\frac{dy}{dx} = x^x (\ln x + 1)}. \]