Question

If \(y = x^x\), then \(\frac{dy}{dx}\) = ____.
 

• A. $x^x$
• B. $x^x(1 + \ln x)$
• C. $x^{x-1}$
• D. $\ln x$

Hint:

Never use the power rule ($nx^{n-1}$) when the exponent is a variable. That rule only applies when the exponent is a constant number.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
To differentiate a function where both the base and the exponent are variables, we use logarithmic differentiation. This allows us to bring the exponent down using log properties.

Step 2: Key Formula or Approach:
1. Take natural logarithm ($\ln$) on both sides. 2. Use the property: $\ln(a^b) = b \ln a$. 3. Differentiate implicitly.

Step 3: Detailed Explanation:
Given $y = x^x$: 1. Taking $\ln$ on both sides: \[ \ln y = \ln(x^x) \implies \ln y = x \ln x \] 2. Differentiating with respect to $x$ using the product rule on the right: \[ \frac{1}{y} \frac{dy}{dx} = (x \cdot \frac{1}{x} + \ln x \cdot 1) \] \[ \frac{1}{y} \frac{dy}{dx} = 1 + \ln x \] 3. Multiply by $y$ to solve for $\frac{dy}{dx}$: \[ \frac{dy}{dx} = y(1 + \ln x) \] 4. Substitute $y = x^x$ back into the equation: \[ \frac{dy}{dx} = x^x(1 + \ln x) \]

Step 4: Final Answer:
The derivative is $x^x(1 + \ln x)$.