If $f(x)=\frac{6 x+5}{7 x+6}$, , find: \[ f^{\prime}(x)= \] \[ f^{\prime}(2)= \]
Solution
Step 1 ：Let's find the derivative of the function \(f(x)=\frac{6 x+5}{7 x+6}\). We can use the quotient rule, which states that the derivative of \(\frac{u}{v}\) is \(\frac{vu' - uv'}{v^2}\), where \(u\) and \(v\) are functions of \(x\), and \(u'\) and \(v'\) are their respective derivatives. In this case, \(u = 6x + 5\) and \(v = 7x + 6\).
Step 2 ：The derivative of \(u\) with respect to \(x\) is \(u' = 6\) and the derivative of \(v\) with respect to \(x\) is \(v' = 7\).
Step 3 ：Substituting these values into the quotient rule gives us \(f'(x) = \frac{v*u' - u*v'}{v^2} = \frac{1}{(7x + 6)^2}\).
Step 4 ：Now, we need to find the value of the derivative at \(x = 2\). Substituting \(x = 2\) into \(f'(x)\) gives us \(f'(2) = \frac{1}{400}\).
Step 5 ：Final Answer: The derivative of the function \(f(x)=\frac{6 x+5}{7 x+6}\) is \(f^{\prime}(x)=\boxed{\frac{1}{(7x + 6)^2}}\) and the value of the derivative at \(x = 2\) is \(f^{\prime}(2)=\boxed{\frac{1}{400}}\).
