Differentiate the following w.r.t. x or t or u as the case may be:
`y = (x^5 - x +2)/( x^3 + 7)`

Answer

To differentiate the function \( y = \frac{x^5 - x + 2}{x^3 + 7} \) with respect to \( x \), we will use the quotient rule for differentiation. The quotient rule states that if you have a function in the form \( \frac{u}{v} \), then the derivative is given by:

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

where \( u \) is the numerator and \( v \) is the denominator.

### Step 1: Identify \( u \) and \( v \)
Let:
- \( u = x^5 - x + 2 \)
- \( v = x^3 + 7 \)

### Step 2: Differentiate \( u \) and \( v \)
Now, we need to find \( \frac{du}{dx} \) and \( \frac{dv}{dx} \).

1. Differentiate \( u \):
\[ \frac{du}{dx} = \frac{d}{dx}(x^5 - x + 2) = 5x^4 - 1 \]

2. Differentiate \( v \):
\[ \frac{dv}{dx} = \frac{d}{dx}(x^3 + 7) = 3x^2 \]

### Step 3: Apply the Quotient Rule
Now we can apply the quotient rule:

\[ \frac{dy}{dx} = \frac{(x^3 + 7)(5x^4 - 1) - (x^5 - x + 2)(3x^2)}{(x^3 + 7)^2} \]

### Step 4: Simplify the Expression
Now we will simplify the numerator:

1. Expand \( (x^3 + 7)(5x^4 - 1) \):
\[ = 5x^7 + 35x^4 - x^3 - 7 \]

2. Expand \( (x^5 - x + 2)(3x^2) \):
\[ = 3x^7 - 3x^3 + 6x^2 \]

3. Combine the results:
\[ \text{Numerator} = (5x^7 + 35x^4 - x^3 - 7) - (3x^7 - 3x^3 + 6x^2) \]
\[ = 5x^7 - 3x^7 + 35x^4 - x^3 + 3x^3 - 6x^2 - 7 \]
\[ = 2x^7 + 35x^4 + 2x^3 - 6x^2 - 7 \]

### Final Result
Thus, the derivative is:

\[ \frac{dy}{dx} = \frac{2x^7 + 35x^4 + 2x^3 - 6x^2 - 7}{(x^3 + 7)^2} \]