Find derivative of given functions w.r.t. the independent variable x.
`y=(x^(2)+1)(x+5+1/x)`

Options

A. `2x(x+5+1/x)+(x^(2)+1)(1-(1)/(x^(2)))`
B. `2x(x+5+1/x)-(x^(2)+1)(1-(1)/(x^(2)))`
C. `2x+1-(1)/(x^(2))`
D. 0

Answer

To find the derivative of the function \( y = (x^2 + 1)(x + 5 + \frac{1}{x}) \) with respect to \( x \), we will apply the product rule of differentiation.

### Step-by-Step Solution

1. **Identify the Functions**:
We can identify two functions in the product:
- \( u = x^2 + 1 \)
- \( v = x + 5 + \frac{1}{x} \)

2. **Differentiate Each Function**:
- The derivative of \( u \) with respect to \( x \):
\[ \frac{du}{dx} = \frac{d}{dx}(x^2 + 1) = 2x \]
- The derivative of \( v \) with respect to \( x \):
\[ \frac{dv}{dx} = \frac{d}{dx}(x + 5 + \frac{1}{x}) = 1 + 0 - \frac{1}{x^2} = 1 - \frac{1}{x^2} \]

3. **Apply the Product Rule**:
The product rule states that if \( y = uv \), then:
\[ \frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx} \]
Substituting the values we found:
\[ \frac{dy}{dx} = (x^2 + 1)(1 - \frac{1}{x^2}) + (x + 5 + \frac{1}{x})(2x) \]

4. **Simplify Each Term**:
- For the first term:
\[ (x^2 + 1)(1 - \frac{1}{x^2}) = (x^2 + 1) - (x^2 + 1) \cdot \frac{1}{x^2} = x^2 + 1 - 1 - \frac{1}{x^2} = x^2 - \frac{1}{x^2} \]
- For the second term:
\[ (x + 5 + \frac{1}{x})(2x) = 2x^2 + 10x + 2 \]

5. **Combine the Results**:
Now, we combine the results from both terms:
\[ \frac{dy}{dx} = (x^2 - \frac{1}{x^2}) + (2x^2 + 10x + 2) \]
Simplifying this gives:
\[ \frac{dy}{dx} = x^2 + 2x^2 + 10x + 2 - \frac{1}{x^2} = 3x^2 + 10x + 2 - \frac{1}{x^2} \]

### Final Result:
Thus, the derivative of the function \( y \) with respect to \( x \) is:
\[ \frac{dy}{dx} = 3x^2 + 10x + 2 - \frac{1}{x^2} \]