find $$ \frac{\differentialD y}{\differentialD x}\text{ ,if y}=x^{x} $$

To find $ \frac{\differentialD y}{\differentialD x} $ where $ y = x^x $, we need to use logarithmic differentiation, which helps in differentiating functions where both the base and the exponent are variable (as in $ x^x $).

### Step 1: Take the natural logarithm of both sides.

\[
y = x^x
\]

Take 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 2: Differentiate both sides with respect to $ x $.

Now, we differentiate both sides with respect to $ x $, remembering that $ y $ is a function of $ x $.

For the left-hand side, use the chain rule:

\[
\frac{d}{dx} [\ln(y)] = \frac{1}{y} \frac{dy}{dx}
\]

For the right-hand side, we differentiate $ x \ln(x) $ using the product rule:

\[
\frac{d}{dx} [x \ln(x)] = \ln(x) + 1
\]

Thus, we now have the equation:

\[
\frac{1}{y} \frac{dy}{dx} = \ln(x) + 1
\]

### Step 3: Solve for $ \frac{dy}{dx} $.

Now, multiply both sides of the equation by $ y $ to isolate $ \frac{dy}{dx} $:

\[
\frac{dy}{dx} = y (\ln(x) + 1)
\]

Since $ y = x^x $, substitute this back in:

\[
\frac{dy}{dx} = x^x (\ln(x) + 1)
\]

### Final Answer:
\[
\frac{dy}{dx} = x^x (\ln(x) + 1)
\]

---

Would you like more details or explanations on any part of this process?

Here are some related questions that could help deepen your understanding:
1. What is logarithmic differentiation and when is it used?
2. How do you differentiate $ x^x $ without logarithmic differentiation?
3. Can you apply this method to differentiate other functions with variable exponents, like $ x^{x^2} $?
4. What is the chain rule and how is it applied in differentiation?
5. How do you handle derivatives of functions with both the base and the exponent as variables?

**Tip:** Logarithmic differentiation is a powerful technique for simplifying the differentiation of functions where the exponent is a function of the base, or vice versa. Always take the log first!

Learn how to differentiate the function y = x^x using logarithmic differentiation. This problem covers key differentiation techniques such as logarithmic differentiation, the chain rule, and the product rule. Suitable for students in Grades 11-12.

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