Find the derivatives of the following :
`g(x)=(3x^(2)-2)/(x^(2)+7)`

Answer

To find the derivative of the function \( g(x) = \frac{3x^2 - 2}{x^2 + 7} \), we will use the quotient rule. The quotient rule states that if you have a function in the form \( \frac{u}{v} \), then the derivative \( g'(x) \) is given by:

\[ g'(x) = \frac{u'v - uv'}{v^2} \]

where \( u = 3x^2 - 2 \) and \( v = x^2 + 7 \).

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

### Step 2: Find the derivatives \( u' \) and \( v' \)
- To find \( u' \):
\[ u' = \frac{d}{dx}(3x^2 - 2) = 6x \]
- To find \( v' \):
\[ v' = \frac{d}{dx}(x^2 + 7) = 2x \]

### Step 3: Apply the quotient rule
Now we substitute \( u \), \( u' \), \( v \), and \( v' \) into the quotient rule formula:
\[ g'(x) = \frac{(6x)(x^2 + 7) - (3x^2 - 2)(2x)}{(x^2 + 7)^2} \]

### Step 4: Simplify the numerator
Now we simplify the numerator:
1. Expand \( (6x)(x^2 + 7) \):
\[ 6x^3 + 42x \]
2. Expand \( (3x^2 - 2)(2x) \):
\[ 6x^3 - 4x \]
3. Substitute these back into the equation:
\[ g'(x) = \frac{(6x^3 + 42x) - (6x^3 - 4x)}{(x^2 + 7)^2} \]
4. Combine like terms:
\[ g'(x) = \frac{6x^3 + 42x - 6x^3 + 4x}{(x^2 + 7)^2} \]
This simplifies to:
\[ g'(x) = \frac{46x}{(x^2 + 7)^2} \]

### Final Result
Thus, the derivative of the function \( g(x) \) is:
\[ g'(x) = \frac{46x}{(x^2 + 7)^2} \]