Question

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

Answer 1

Explanation:

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

`(dy)/(dx) = (v(du)/(dx) - u(dv)/(dx))/(v^2)\:\:\:\:\:\:\:\:\:(1)`

We can see that

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

`(du)/(dx) = (8x)/(4x^2 + 1)`

`v(x) = 2x - 3`

`(dv)/(dx) = 2`

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

`(dy)/(dx) = ((8x(2x - 3))/((4x^2 + 1)) - 2\ln (4x^2 + 1))/((2x - 3)^2)`

`\:\:\:\:\:\:\:= (8x)/((4x^2 + 1)(2x - 3)) - (2\ln (4x^2 + 1))/((2x - 3)^2)`