Question

Differentiate \( X^X \) with respect to \( X \).

Hint:

To differentiate \( X^X \), use logarithmic differentiation. It simplifies the process of differentiating functions with variable exponents.

Answer 1

Step 1: Understand the function.
The function is \( X^X \), which is a power function where both the base and the exponent are \( X \). To differentiate this, we will use logarithmic differentiation.

Step 2: Apply logarithmic differentiation.
Let \( y = X^X \). Taking the natural logarithm of both sides: \[ \ln y = \ln(X^X) \] Using the logarithmic property \( \ln(a^b) = b \ln a \), we get: \[ \ln y = X \ln X \]

Step 3: Differentiate both sides.
Now differentiate both sides with respect to \( X \): \[ \frac{d}{dX}(\ln y) = \frac{d}{dX}(X \ln X) \] On the left-hand side, by the chain rule: \[ \frac{1}{y} \frac{dy}{dX} \] On the right-hand side, use the product rule: \[ \frac{d}{dX}(X \ln X) = \ln X + 1 \]

Step 4: Solve for \( \frac{dy}{dX} \).
Now, multiply both sides by \( y \) to get: \[ \frac{dy}{dX} = X^X (\ln X + 1) \]

Step 5: Conclusion.
Thus, the derivative of \( X^X \) with respect to \( X \) is: \[ \frac{dy}{dX} = X^X (\ln X + 1) \]