Step-by-step explanation:
According to the quotient rule of differentiation, if [tex]y(x) = u(x)/v(x),[/tex] then its derivative is given by
[tex]\dfrac{dy}{dx} = \dfrac{v\dfrac{du}{dx} - u\dfrac{dv}{dx}}{v^2}\:\:\:\:\:\:\:\:\:(1)[/tex]
We can see that
[tex]u(x) = \ln (4x^2 + 1)[/tex]
[tex]\dfrac{du}{dx} = \dfrac{8x}{4x^2 + 1}[/tex]
[tex]v(x) = 2x - 3[/tex]
[tex]\dfrac{dv}{dx} = 2[/tex]
Plugging the above expressions into Eqn(1), we find that the derivative is
[tex]\dfrac{dy}{dx} = \dfrac{\dfrac{8x(2x - 3)}{(4x^2 + 1)} - 2\ln (4x^2 + 1)}{(2x - 3)^2}[/tex]
[tex]\:\:\:\:\:\:\:= \dfrac{8x}{(4x^2 + 1)(2x - 3)} - \dfrac{2\ln (4x^2 + 1)}{(2x - 3)^2}[/tex]
