If `y = 1/x` , then the value of `(dy)/(sqrt(1 + y^4)) + (dx)/(sqrt(1 + x^4)) + 3` is equal to

Options

A. 0
B. 3
C. 4
D. `-3`

Answer

To solve the problem, we start with the given equation:

\[ y = \frac{1}{x} \]

We need to find the value of the expression:

\[ \frac{dy}{\sqrt{1 + y^4}} + \frac{dx}{\sqrt{1 + x^4}} + 3 \]

### Step 1: Differentiate \( y \)

First, we differentiate \( y \) with respect to \( x \):

\[ \frac{dy}{dx} = \frac{d}{dx}\left(\frac{1}{x}\right) = -\frac{1}{x^2} \]

### Step 2: Substitute \( dy \)

Now, we can express \( dy \) in terms of \( dx \):

\[ dy = -\frac{1}{x^2} dx \]

### Step 3: Substitute \( y \) into the expression

Next, we substitute \( y = \frac{1}{x} \) into the expression:

\[ \frac{dy}{\sqrt{1 + y^4}} = \frac{-\frac{1}{x^2} dx}{\sqrt{1 + \left(\frac{1}{x}\right)^4}} = \frac{-\frac{1}{x^2} dx}{\sqrt{1 + \frac{1}{x^4}}} \]

### Step 4: Simplify the square root

Now, simplify the square root:

\[ \sqrt{1 + \frac{1}{x^4}} = \sqrt{\frac{x^4 + 1}{x^4}} = \frac{\sqrt{x^4 + 1}}{x^2} \]

### Step 5: Substitute back into the expression

Substituting this back into our expression gives:

\[ \frac{-\frac{1}{x^2} dx}{\frac{\sqrt{x^4 + 1}}{x^2}} = \frac{-dx}{\sqrt{x^4 + 1}} \]

### Step 6: Substitute \( dx \)

Now, substitute this into the original expression:

\[ -\frac{dx}{\sqrt{x^4 + 1}} + \frac{dx}{\sqrt{1 + x^4}} + 3 \]

Notice that the first two terms are equal and opposite:

\[ -\frac{dx}{\sqrt{x^4 + 1}} + \frac{dx}{\sqrt{x^4 + 1}} = 0 \]

### Step 7: Final result

Thus, we are left with:

\[ 0 + 3 = 3 \]

Therefore, the value of the expression is:

\[ \boxed{3} \]