Chapter 4.3, Problem 51E

Katie and Sarah are working on taking the derivative of

$f\left(x\right)=\frac{2x}{3x+4}.$

Margy uses the quotient rule and chain rule as follows:

$f^{\text{'}}\left(x\right)=\frac{\left(3x+4\right)2-2x\left(3\right)}{{\left(3x+4\right)}^2}=\frac{8}{{\left(3x+4\right)}^2}.$

Sarah converts it into a product and uses the product rule and the chain rule:

$f\left(x\right)=2x{\left(3x+4\right)}^{-1}$

$f^{\text{'}}\left(x\right)=2x\left(-1\right){\left(3x+4\right)}^{-2}\left(3\right)+2{\left(3x+4\right)}^{-1}$

$=2x{\left(3x+4\right)}^{-1}-6x{\left(3x+4\right)}^{-2}.$

Explain the discrepancies between the two answers. Which procedure do you think is preferable?