Final answer:
To find the derivative D_(x)y of the function y=(4x²+7x+8)/(2x+9), apply the quotient rule for differentiation which involves calculating the derivatives of the numerator and the denominator separately and then combining them as per the rule.
Step-by-step explanation:
The question asks to find the derivative of the function y=(4x²+7x+8)/(2x+9) with respect to x, denoted as D_(x)y or y'. This calculation requires the application of the quotient rule for differentiation. The quotient rule states that the derivative of a function that is the quotient of two functions, u(x) over v(x), is given by (v(x)u'(x) - u(x)v'(x)) / (v(x))^2. Applying this rule to the given function:
- Let u(x) = 4x²+7x+8 and calculate its derivative u'(x) = 8x+7.
- Let v(x) = 2x+9 and calculate its derivative v'(x) = 2.
- Apply the quotient rule: y' = ((2x+9)(8x+7) - (4x²+7x+8)(2)) / (2x+9)^2.
To fully answer the question, you would proceed to simplify the resulting expression to find the derivative in its simplest forms.