Find `dy/dx if y=x^x`

Answer

To find \(\frac{dy}{dx}\) for the function \(y = x^x\), we will follow these steps:

### Step 1: Take the logarithm of both sides
We start with the equation:
\[ y = x^x \]
Taking the natural logarithm on both sides gives us:
\[ \log y = \log(x^x) \]

### Step 2: Apply the logarithmic property
Using the property of logarithms that states \(\log(a^b) = b \log(a)\), we can rewrite the equation:
\[ \log y = x \log x \]

### Step 3: Differentiate both sides
Now, we differentiate both sides with respect to \(x\). We will use implicit differentiation on the left side and the product rule on the right side:
\[ \frac{d}{dx}(\log y) = \frac{d}{dx}(x \log x) \]
Using the chain rule on the left side:
\[ \frac{1}{y} \frac{dy}{dx} = \frac{d}{dx}(x \log x) \]

### Step 4: Differentiate the right side using the product rule
For the right side, we apply the product rule:
\[ \frac{d}{dx}(x \log x) = \frac{d}{dx}(x) \cdot \log x + x \cdot \frac{d}{dx}(\log x) \]
This simplifies to:
\[ 1 \cdot \log x + x \cdot \frac{1}{x} = \log x + 1 \]

### Step 5: Combine the results
Now we have:
\[ \frac{1}{y} \frac{dy}{dx} = \log x + 1 \]
Multiplying both sides by \(y\) gives us:
\[ \frac{dy}{dx} = y(\log x + 1) \]

### Step 6: Substitute back for \(y\)
Since \(y = x^x\), we substitute back:
\[ \frac{dy}{dx} = x^x(\log x + 1) \]

### Final Answer
Thus, the derivative \(\frac{dy}{dx}\) is:
\[ \frac{dy}{dx} = x^x(1 + \log x) \]

---