Questions: Find the derivative of the function. g(x)=(7+5)/(2+6x)

Solution Steps

To find the derivative of the function \( g(x) = \frac{7+5}{2+6x} \), we first simplify the function and then apply the quotient rule. The quotient rule states that if you have a function \( \frac{u(x)}{v(x)} \), its derivative is given by \( \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} \). Here, \( u(x) = 12 \) and \( v(x) = 2 + 6x \).

We start with the function given by

\[ g(x) = \frac{7 + 5}{2 + 6x} = \frac{12}{6x + 2}. \]

To find the derivative \( g'(x) \), we apply the quotient rule, which states:

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

where \( u = 12 \) and \( v = 6x + 2 \).

We compute the derivatives:

• \( u' = 0 \) (since \( u \) is a constant),
• \( v' = 6 \).

Substituting into the quotient rule gives:

\[ g'(x) = \frac{0 \cdot (6x + 2) - 12 \cdot 6}{(6x + 2)^2} = \frac{-72}{(6x + 2)^2}. \]

Final Answer