Differentiate the following w.r.t. x :
`(cosx)^(x)+(sinx)^(1//x)`

Answer

To differentiate the function \( y = (\cos x)^x + (\sin x)^{\frac{1}{x}} \) with respect to \( x \), we will use the properties of logarithms and the product rule for differentiation. Let's break it down step by step.

### Step 1: Differentiate \( u = (\cos x)^x \)

1. **Take the natural logarithm of both sides:**
\[ \log u = x \log (\cos x) \]

2. **Differentiate both sides with respect to \( x \):**
Using implicit differentiation:
\[ \frac{1}{u} \frac{du}{dx} = \log (\cos x) + x \cdot \frac{-\sin x}{\cos x} \]
Here, we used the product rule on \( x \log (\cos x) \).

3. **Solve for \( \frac{du}{dx} \):**
\[ \frac{du}{dx} = u \left( \log (\cos x) - x \tan x \right) \]
Substituting back \( u = (\cos x)^x \):
\[ \frac{du}{dx} = (\cos x)^x \left( \log (\cos x) - x \tan x \right) \]

### Step 2: Differentiate \( v = (\sin x)^{\frac{1}{x}} \)

1. **Take the natural logarithm of both sides:**
\[ \log v = \frac{1}{x} \log (\sin x) \]

2. **Differentiate both sides with respect to \( x \):**
Using the quotient rule:
\[ \frac{1}{v} \frac{dv}{dx} = \frac{-1}{x^2} \log (\sin x) + \frac{1}{x} \cdot \frac{\cos x}{\sin x} \]

3. **Solve for \( \frac{dv}{dx} \):**
\[ \frac{dv}{dx} = v \left( \frac{\cos x}{\sin x} \cdot \frac{1}{x} - \frac{\log (\sin x)}{x^2} \right) \]
Substituting back \( v = (\sin x)^{\frac{1}{x}} \):
\[ \frac{dv}{dx} = (\sin x)^{\frac{1}{x}} \left( \frac{\cot x}{x} - \frac{\log (\sin x)}{x^2} \right) \]

### Step 3: Combine the derivatives

Now, we can combine the derivatives of \( u \) and \( v \) to find \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{du}{dx} + \frac{dv}{dx} \]
Substituting the expressions we found:
\[ \frac{dy}{dx} = (\cos x)^x \left( \log (\cos x) - x \tan x \right) + (\sin x)^{\frac{1}{x}} \left( \frac{\cot x}{x} - \frac{\log (\sin x)}{x^2} \right) \]

### Final Answer
Thus, the derivative of \( y \) with respect to \( x \) is:
\[ \frac{dy}{dx} = (\cos x)^x \left( \log (\cos x) - x \tan x \right) + (\sin x)^{\frac{1}{x}} \left( \frac{\cot x}{x} - \frac{\log (\sin x)}{x^2} \right) \]