Differentiate the following w.r.t. x :
`x^(cotx)+(cosx)^(sinx)`

Answer

To differentiate the function \( y = x^{\cot x} + (\cos x)^{\sin x} \) with respect to \( x \), we will follow these steps:

### Step 1: Rewrite the function
Let \( y = x^{\cot x} + (\cos x)^{\sin x} \).

### Step 2: Differentiate each term separately
We will differentiate \( y \) term by term.

#### For the first term \( x^{\cot x} \):
Using the property \( a^b = e^{b \ln a} \), we can rewrite:
\[ x^{\cot x} = e^{\cot x \ln x} \]
Now, differentiate using the chain rule:
\[ \frac{dy_1}{dx} = e^{\cot x \ln x} \cdot \frac{d}{dx}(\cot x \ln x) \]

Using the product rule for \( \cot x \ln x \):
\[ \frac{d}{dx}(\cot x \ln x) = \cot x \cdot \frac{d}{dx}(\ln x) + \ln x \cdot \frac{d}{dx}(\cot x) \]
We know:
\[ \frac{d}{dx}(\ln x) = \frac{1}{x}, \quad \text{and} \quad \frac{d}{dx}(\cot x) = -\csc^2 x \]
Thus,
\[ \frac{d}{dx}(\cot x \ln x) = \cot x \cdot \frac{1}{x} - \ln x \cdot \csc^2 x \]
So,
\[ \frac{dy_1}{dx} = x^{\cot x} \left( \frac{\cot x}{x} - \ln x \cdot \csc^2 x \right) \]

#### For the second term \( (\cos x)^{\sin x} \):
Again, using the property \( a^b = e^{b \ln a} \):
\[ (\cos x)^{\sin x} = e^{\sin x \ln(\cos x)} \]
Now, differentiate using the chain rule:
\[ \frac{dy_2}{dx} = e^{\sin x \ln(\cos x)} \cdot \frac{d}{dx}(\sin x \ln(\cos x)) \]

Using the product rule for \( \sin x \ln(\cos x) \):
\[ \frac{d}{dx}(\sin x \ln(\cos x)) = \sin x \cdot \frac{d}{dx}(\ln(\cos x)) + \ln(\cos x) \cdot \frac{d}{dx}(\sin x) \]
We know:
\[ \frac{d}{dx}(\ln(\cos x)) = -\tan x, \quad \text{and} \quad \frac{d}{dx}(\sin x) = \cos x \]
Thus,
\[ \frac{d}{dx}(\sin x \ln(\cos x)) = \sin x (-\tan x) + \ln(\cos x) \cos x \]
So,
\[ \frac{dy_2}{dx} = (\cos x)^{\sin x} \left( -\sin x \tan x + \ln(\cos x) \cos x \right) \]

### Step 3: Combine the derivatives
Now we can combine the derivatives of both terms:
\[ \frac{dy}{dx} = \frac{dy_1}{dx} + \frac{dy_2}{dx} \]
Substituting the expressions we derived:
\[ \frac{dy}{dx} = x^{\cot x} \left( \frac{\cot x}{x} - \ln x \cdot \csc^2 x \right) + (\cos x)^{\sin x} \left( -\sin x \tan x + \ln(\cos x) \cos x \right) \]

### Final Answer
Thus, the derivative of \( y \) with respect to \( x \) is:
\[ \frac{dy}{dx} = x^{\cot x} \left( \frac{\cot x}{x} - \ln x \cdot \csc^2 x \right) + (\cos x)^{\sin x} \left( -\sin x \tan x + \ln(\cos x) \cos x \right) \]