Differentiate the following w.r.t. x :
`(logx)^(cosx)+(x^(2)+1)/(x^(2)-1)`

Answer

To differentiate the given function \( y = (\log x)^{\cos x} + \frac{x^2 + 1}{x^2 - 1} \) with respect to \( x \), we will follow these steps:

### Step 1: Identify the components of the function
The function can be split into two parts:
- \( u = (\log x)^{\cos x} \)
- \( v = \frac{x^2 + 1}{x^2 - 1} \)

Thus, we can express \( y \) as:
\[ y = u + v \]

### Step 2: Differentiate using the sum rule
Using the sum rule of differentiation, we have:
\[ \frac{dy}{dx} = \frac{du}{dx} + \frac{dv}{dx} \]

### Step 3: Differentiate \( u = (\log x)^{\cos x} \)
To differentiate \( u \), we will use logarithmic differentiation:
1. Take the natural logarithm of both sides:
\[ \log u = \cos x \cdot \log(\log x) \]
2. Differentiate both sides with respect to \( x \):
\[ \frac{1}{u} \frac{du}{dx} = -\sin x \cdot \log(\log x) + \cos x \cdot \frac{1}{\log x} \cdot \frac{1}{x} \]
3. Rearranging gives:
\[ \frac{du}{dx} = u \left( -\sin x \cdot \log(\log x) + \frac{\cos x}{x \log x} \right) \]
4. Substitute back \( u = (\log x)^{\cos x} \):
\[ \frac{du}{dx} = (\log x)^{\cos x} \left( -\sin x \cdot \log(\log x) + \frac{\cos x}{x \log x} \right) \]

### Step 4: Differentiate \( v = \frac{x^2 + 1}{x^2 - 1} \)
Using the quotient rule:
1. Let \( a = x^2 + 1 \) and \( b = x^2 - 1 \).
2. The derivative using the quotient rule is:
\[ \frac{dv}{dx} = \frac{b \cdot \frac{da}{dx} - a \cdot \frac{db}{dx}}{b^2} \]
3. Calculate \( \frac{da}{dx} = 2x \) and \( \frac{db}{dx} = 2x \):
\[ \frac{dv}{dx} = \frac{(x^2 - 1)(2x) - (x^2 + 1)(2x)}{(x^2 - 1)^2} \]
4. Simplifying gives:
\[ \frac{dv}{dx} = \frac{2x(x^2 - 1 - x^2 - 1)}{(x^2 - 1)^2} = \frac{-4x}{(x^2 - 1)^2} \]

### Step 5: Combine the derivatives
Now we can combine the derivatives:
\[ \frac{dy}{dx} = \frac{du}{dx} + \frac{dv}{dx} \]
Substituting the expressions we found:
\[ \frac{dy}{dx} = (\log x)^{\cos x} \left( -\sin x \cdot \log(\log x) + \frac{\cos x}{x \log x} \right) - \frac{4x}{(x^2 - 1)^2} \]

### Final Answer
Thus, the derivative of the given function is:
\[ \frac{dy}{dx} = (\log x)^{\cos x} \left( -\sin x \cdot \log(\log x) + \frac{\cos x}{x \log x} \right) - \frac{4x}{(x^2 - 1)^2} \]