Differentiate `(xcosx)^x` with respect to `xdot`

Explanation

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

### Step 1: Take the natural logarithm of both sides
We start by taking the logarithm of both sides to simplify the differentiation process:
\[ \log y = \log((x \cos x)^x) \]

### Step 2: Apply logarithmic properties
Using the properties of logarithms, we can rewrite the right side:
\[ \log y = x \log(x \cos x) \]
Next, we can further expand this using the product property of logarithms:
\[ \log y = x (\log x + \log(\cos x)) \]
Thus, we have:
\[ \log y = x \log x + x \log(\cos x) \]

### Step 3: Differentiate both sides with respect to \( x \)
Now we differentiate both sides with respect to \( x \):
\[ \frac{1}{y} \frac{dy}{dx} = \frac{d}{dx}(x \log x) + \frac{d}{dx}(x \log(\cos x)) \]

### Step 4: Differentiate the first term
Using the product rule on \( x \log x \):
\[ \frac{d}{dx}(x \log x) = \log x + 1 \]

### Step 5: Differentiate the second term
Now, we apply the product rule to \( x \log(\cos x) \):
\[ \frac{d}{dx}(x \log(\cos x)) = \log(\cos x) + x \frac{d}{dx}(\log(\cos x)) \]
The derivative of \( \log(\cos x) \) is:
\[ \frac{d}{dx}(\log(\cos x)) = -\tan x \]
Thus,
\[ \frac{d}{dx}(x \log(\cos x)) = \log(\cos x) - x \tan x \]

### Step 6: Combine the results
Now we can combine the derivatives:
\[ \frac{1}{y} \frac{dy}{dx} = (\log x + 1) + (\log(\cos x) - x \tan x) \]
This simplifies to:
\[ \frac{1}{y} \frac{dy}{dx} = \log x + \log(\cos x) + 1 - x \tan x \]

### Step 7: Solve for \( \frac{dy}{dx} \)
Now, we multiply both sides by \( y \) to isolate \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = y \left( \log x + \log(\cos x) + 1 - x \tan x \right) \]

### Step 8: Substitute back for \( y \)
Recall that \( y = (x \cos x)^x \):
\[ \frac{dy}{dx} = (x \cos x)^x \left( \log x + \log(\cos x) + 1 - x \tan x \right) \]

### Final Result
Thus, the derivative of \( (x \cos x)^x \) with respect to \( x \) is:
\[ \frac{dy}{dx} = (x \cos x)^x \left( \log x + \log(\cos x) + 1 - x \tan x \right) \]