Question

Refer to image given and please answer!!!!!

Answer 1

Step-by-step explanation:

According to the quotient rule of differentiation, if $y(x) = u(x)/v(x),$ then its derivative is given by

$\dfrac{dy}{dx} = \dfrac{v\dfrac{du}{dx} - u\dfrac{dv}{dx}}{v^2}\:\:\:\:\:\:\:\:\:(1)$

We can see that

$u(x) = \ln (4x^2 + 1)$

$\dfrac{du}{dx} = \dfrac{8x}{4x^2 + 1}$

$v(x) = 2x - 3$

$\dfrac{dv}{dx} = 2$

Plugging the above expressions into Eqn(1), we find that the derivative is

$\dfrac{dy}{dx} = \dfrac{\dfrac{8x(2x - 3)}{(4x^2 + 1)} - 2\ln (4x^2 + 1)}{(2x - 3)^2}$

$\:\:\:\:\:\:\:= \dfrac{8x}{(4x^2 + 1)(2x - 3)} - \dfrac{2\ln (4x^2 + 1)}{(2x - 3)^2}$