Differentiate the following w.r.t. x :
`(x^(x))^(x)`

Answer

To differentiate the function \( (x^x)^x \) with respect to \( x \), we can follow these steps:

### Step 1: Rewrite the function
Let \( y = (x^x)^x \). We can simplify this using the properties of exponents:
\[ y = x^{x^2} \]

### Step 2: Take the natural logarithm of both sides
To make differentiation easier, we take the natural logarithm of both sides:
\[ \ln y = \ln(x^{x^2}) \]

### Step 3: Apply the logarithmic identity
Using the property of logarithms that states \( \ln(a^b) = b \ln a \), we can rewrite the right side:
\[ \ln y = x^2 \ln x \]

### Step 4: 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:
\[ \frac{d}{dx}(\ln y) = \frac{1}{y} \frac{dy}{dx} \]
For the right side, applying the product rule:
\[ \frac{d}{dx}(x^2 \ln x) = 2x \ln x + x^2 \cdot \frac{1}{x} = 2x \ln x + x \]

### Step 5: Set the derivatives equal
Now we set the derivatives equal to each other:
\[ \frac{1}{y} \frac{dy}{dx} = 2x \ln x + x \]

### Step 6: Solve for \( \frac{dy}{dx} \)
Multiplying both sides by \( y \):
\[ \frac{dy}{dx} = y(2x \ln x + x) \]

### Step 7: Substitute back for \( y \)
Recall that \( y = (x^x)^x = x^{x^2} \):
\[ \frac{dy}{dx} = x^{x^2}(2x \ln x + x) \]

### Final Result
Thus, the derivative of \( (x^x)^x \) with respect to \( x \) is:
\[ \frac{dy}{dx} = x^{x^2}(2x \ln x + x) \]

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