Question

1. if ( y=\frac{2 x+3}{3 x+2} ), then ( \frac{d y}{d x}= )

a ( \frac{12 x+13}{(3 x+2)^{2}} )
b ( \frac{12 x-13}{(3 x+2)^{2}} )
c ( \frac{5}{(3 x+2)^{2}} )
d ( \frac{-5}{(3 x+2)^{2}} )
e ( \frac{2}{3} )

Explanation:

Step1: Apply the quotient rule

The quotient rule states that if \(y=\frac{u}{v}\), then \(\frac{dy}{dx}=\frac{u'v - uv'}{v^{2}}\). Here, \(u = 2x + 3\), so \(u'=2\); \(v=3x + 2\), so \(v'=3\).

Step2: Substitute into the quotient rule formula

$$
LATEXBLOCK0
$$

Answer:

D. \(\frac{-5}{(3x + 2)^{2}}\)