Find the derivative of y w.r.t. x in each of the following :
`y/(x+y)=3+x^(3)`.

Answer

To find the derivative of \( y \) with respect to \( x \) in the equation

\[ \frac{y}{x+y} = 3 + x^3, \]

we will follow these steps:

### Step 1: Cross-multiply the equation
We start by cross-multiplying to eliminate the fraction:

\[ y = (3 + x^3)(x + y). \]

### Step 2: Expand the equation
Now, we expand the right-hand side:

\[ y = (3 + x^3)x + (3 + x^3)y. \]

This simplifies to:

\[ y = 3x + x^4 + 3y + x^3y. \]

### Step 3: Rearrange the equation
Next, we rearrange the equation to group all terms involving \( y \) on one side:

\[ y - 3y - x^3y = 3x + x^4. \]

This simplifies to:

\[ (1 - 3 - x^3)y = 3x + x^4. \]

### Step 4: Factor out \( y \)
Now we can factor out \( y \):

\[ y(1 - 3 - x^3) = 3x + x^4. \]

### Step 5: Solve for \( y \)
We can express \( y \) in terms of \( x \):

\[ y = \frac{3x + x^4}{1 - 3 - x^3}. \]

### Step 6: Differentiate both sides
Now, we differentiate both sides with respect to \( x \):

Using the quotient rule, where \( u = 3x + x^4 \) and \( v = 1 - 3 - x^3 \):

\[ \frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}. \]

### Step 7: Find \( \frac{du}{dx} \) and \( \frac{dv}{dx} \)
Calculating \( \frac{du}{dx} \):

\[ \frac{du}{dx} = 3 + 4x^3, \]

and calculating \( \frac{dv}{dx} \):

\[ \frac{dv}{dx} = -3x^2. \]

### Step 8: Substitute back into the derivative
Now substituting back into the derivative formula:

\[ \frac{dy}{dx} = \frac{(1 - 3 - x^3)(3 + 4x^3) - (3x + x^4)(-3x^2)}{(1 - 3 - x^3)^2}. \]

### Step 9: Simplify the expression
Now we simplify the numerator:

1. Expand \( (1 - 3 - x^3)(3 + 4x^3) \).
2. Expand \( (3x + x^4)(-3x^2) \).
3. Combine like terms.

### Final Result
After simplification, we will have the derivative \( \frac{dy}{dx} \) expressed in terms of \( x \).