Final answer:
To find the derivative of the function y=(4e^(9x))/(9x+2), the quotient and chain rules of differentiation must be applied. First, one defines u and v as functions of x where y=u/v, then calculates the derivatives u' and v'. Finally, apply these results into the quotient formula to obtain y'.
Step-by-step explanation:
The student is asking for the derivative of the function y=(4e^(9x))/(9x+2). This requires the use of the quotient rule and the chain rule for derivatives. The quotient rule states that the derivative of a function that is the quotient of two other functions, h(x)=f(x)/g(x), is given by h'(x)=[g(x)f'(x)-f(x)g'(x)]/[g(x)]^2. The chain rule is used when taking the derivative of a function composed of other functions, and it states that if h(x) = f(g(x)), then h'(x) is f'(g(x))g'(x).
To find the derivative of y with respect to x, let u=4e^(9x) and v=9x+2 where y=u/v. Applying the quotient rule, we get:
y'=(vu' - uv') / v^2
To find u', we use the chain rule, and v' is simply the derivative of 9x+2 with respect to x. After computing u', uv', and v^2, we combine and simplify for the final derivative expression.