Question

Use the quotient rule to find the derivative of the function: f(x)=9x-5/2x+3.

Answer 1

The derivative of the function f(x) = (9x - 5)/(2x + 3) using the quotient rule is 37/((2x + 3)^2), after applying the rule and simplifying the expression.

Step-by-step explanation:

To find the derivative of the function f(x) = (9x - 5)/(2x + 3) using the quotient rule, we need to follow a specific pattern, where if we have a function in the form of u(x)/v(x), its derivative f'(x) is given by (v(x) * u'(x) - u(x) * v'(x))/(v(x))^2.

Let's apply this to our function:

1. First, identify u(x) as 9x - 5 and v(x) as 2x + 3.
2. Compute u'(x), the derivative of u(x), which is 9.
3. Compute v'(x), the derivative of v(x), which is 2.
4. Apply the quotient rule: f'(x) = ((2x + 3) * 9 - (9x - 5) * 2)/((2x + 3)^2).
5. Simplify the numerator and combine like terms: f'(x) = (18x + 27 - 18x + 10)/((2x + 3)^2), which simplifies to f'(x) = 37/((2x + 3)^2).

Therefore, the derivative of the function f(x) using the quotient rule is 37/((2x + 3)^2).