Question:

Find the derivative of the function $f(x) = \frac{3x^2 - 2x + 5}{2x - 1}$ using the product and quotient rules.

Answer:

To find the derivative of the given function using the product and quotient rules, we can apply the following steps:

Step 1: Start by using the quotient rule, which states that the derivative of a quotient of two functions is given by: [ \frac{d}{dx} \left( \frac{u(x)}{v(x)} \right) = \frac{v(x)u'(x) - u(x)v'(x)}{(v(x))^2} ]

In this case, let $u(x) = 3x^2 - 2x + 5$ and $v(x) = 2x - 1$. Then, we have: [ f'(x) = \frac{(2x - 1)(6x - 2) - (3x^2 - 2x + 5)(2)}{(2x - 1)^2} ]

Step 2: Simplify the numerator and denominator: [ f'(x) = \frac{12x^2 - 4x - 6x + 2 - 6x^2 + 4x - 10}{(2x - 1)^2} ] [ f'(x) = \frac{6x^2 - 14}{(2x - 1)^2} ]

So, the derivative of the function $f(x) = \frac{3x^2 - 2x + 5}{2x - 1}$ using the product and quotient rules is: [ f'(x) = \frac{6x^2 - 14}{(2x - 1)^2} ]

Therefore, the derivative of the given function is $\frac{6x^2 - 14}{(2x - 1)^2}$.