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

Answer

To differentiate the function \( y = (x \cos x)^x \) with respect to \( x \), we can use logarithmic differentiation. Here’s the step-by-step solution:

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

### Step 2: Simplify using logarithmic properties
Using the property of logarithms that states \( \ln(a^b) = b \ln a \), we can simplify:
\[ \ln y = x \ln(x \cos x) \]

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

### Step 4: Apply the product rule
Using the product rule on the right side:
\[ \frac{d}{dx}(x \ln(x \cos x)) = \ln(x \cos x) + x \frac{d}{dx}(\ln(x \cos x)) \]

### Step 5: Differentiate \( \ln(x \cos x) \)
Now, we need to differentiate \( \ln(x \cos x) \):
\[ \frac{d}{dx}(\ln(x \cos x)) = \frac{1}{x \cos x} \cdot \frac{d}{dx}(x \cos x) \]
Using the product rule again for \( x \cos x \):
\[ \frac{d}{dx}(x \cos x) = \cos x - x \sin x \]
So,
\[ \frac{d}{dx}(\ln(x \cos x)) = \frac{\cos x - x \sin x}{x \cos x} \]

### Step 6: Substitute back into the equation
Now we substitute this back into our differentiation equation:
\[ \frac{1}{y} \frac{dy}{dx} = \ln(x \cos x) + x \cdot \frac{\cos x - x \sin x}{x \cos x} \]
This simplifies to:
\[ \frac{1}{y} \frac{dy}{dx} = \ln(x \cos x) + \frac{\cos x - x \sin x}{\cos x} \]

### Step 7: Multiply by \( y \)
Now, multiply both sides by \( y \) to solve for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = y \left( \ln(x \cos x) + \frac{\cos x - x \sin x}{\cos x} \right) \]
Substituting back \( y = (x \cos x)^x \):
\[ \frac{dy}{dx} = (x \cos x)^x \left( \ln(x \cos x) + \frac{\cos x - x \sin x}{\cos x} \right) \]

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