Let's start off with the definition of the quotient rule:
d/dx (f(x)/(g(x))) = (g(x)f'(x) - f(x)g'(x))/(g(x))^2
Or to make remembering the rule simpler, let f(x) be "high," being the upper function and let g(x) be "low," being the lower function to give us:
d/dx ((high)/(low)) = (low*d'(high) - (high*d'(low)))/(low)^2
where d' denotes the "derivative of."
This formula can be read as "low d high minus high d low over low squared," which has a nice flow for memorization.
In this case, let f(x) = (14x+6x^2)/(7+6x)^2 , and let the "low" function denote (7+6x)^2 and the "high" function denote (14x+6x^2)
Applying the quotient rule,
d/dx ((high)/(low)) =
(((7+6x)^2)*d'(14x+6x^2) - ((14x+6x^2)*d'((7+6x)^2)))/((7+6x)^2)^2
Now switching ((high)/(low)) to f(x) for simplicity,
d/dx (f(x)) = (((7+6x)^2)⋅(14 + 12x) - ((14x+6x^2)⋅(2*(7+6x)*6)))/((7+6x)^2)^2
Knowing that by the power rule: d/dx (x^n) = nx^(n-1)
And that by the chain rule: d/dx (f(g(x)) = f'(g(x))⋅g'(x)
And using algebraic simplifications:
d/dx (f(x)) = (((7+6x)^2)⋅2⋅(7 + 6x) - ((14x+6x^2)⋅(84+72x)))/((7+6x)^4)
=(2⋅(7 + 6x)^3 - ((14x+6x^2)⋅(84+72x)))/((7+6x)^4)
=(2⋅(7 + 6x)^3 - ((14x+6x^2)⋅12⋅(7+6x)))/((7+6x)^4)
dividing by (7+6x)
=(2⋅(7 + 6x)^2 - ((14x+6x^2)⋅12))/((7+6x)^3)
and foiling out the expressions, we get:
=(2⋅(49+84x+36x^2) - (168x+72x^2))/((7+6x)^3)
=((98+168x+72x^2) - (168x+72x^2))/((7+6x)^3)
=(98)/((7+6x)^3)