Use brackets to denote arguments of sinusoidal terms such as \( \cos (4 \mathrm{x}) \) to represent \( \cos (4 x) \) as opposed to \( \cos 4 \mathrm{x} \) \( e^{2 x} \) is entered as \( \mathrm{e}^{\wedge}(2 \mathrm{x}) \) not as \( \mathrm{e}^{\wedge} 2 \mathrm{x} \) which would give \( e^{2} x \). a) Use the quotient rule to differentiacte \[ \frac{d y}{d x}=\frac{4 x^{2}-7 x}{6 x+1} \]

To differentiate the function \( y = \frac{4 x^{2}-7 x}{6 x+1} \) using the quotient rule, we start by recalling that the quotient rule states: \[ \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^{2}} \] where \( u = 4 x^{2}-7 x \) and \( v = 6 x+1 \). Now, calculating the derivatives, we find \( u' = 8 x - 7 \) and \( v' = 6 \). Plugging these into the formula gives us: \[ \frac{d y}{d x} = \frac{(8 x - 7)(6 x + 1) - (4 x^{2}-7 x)(6)}{(6 x + 1)^{2}} \] This result simplifies to provide the derivative of the function. Now, once you've differentiated, don’t forget to simplify your final answer to make it easier to work with—you don't want to get lost in the algebraic jungle! Check out each term carefully and factor when you can. Happy differentiating!