Question:

Find the derivative of the function f(x) = (x^2 + 3x - 2)/(2x + 1) using the quotient rule.

Answer:

To find the derivative of the given function using the quotient rule, first recall the quotient rule formula:

The quotient rule states that if f(x) = g(x)/h(x), then f'(x) = (g'(x) * h(x) - g(x) * h'(x))/[h(x)]^2.

Now, let's differentiate the given function step-by-step using the quotient rule:

Step 1: Identify the numerator and denominator functions:

g(x) = x^2 + 3x - 2 (numerator)

h(x) = 2x + 1 (denominator)

Step 2: Differentiate both the numerator and the denominator:

g'(x) = 2x + 3 (derivative of the numerator)

h'(x) = 2 (derivative of the denominator)

Step 3: Plug in the values obtained from differentiation into the quotient rule formula:

f'(x) = [(2x + 3)(2x + 1) - (x^2 + 3x - 2)(2)] / [(2x + 1)^2]

Simplifying the above expression:

f'(x) = (4x^2 + 8x + 6x + 3 - 2x^2 - 6x + 4) / (4x^2 + 4x + 4x + 1)

f'(x) = (2x^2 + 7x + 7) / (4x^2 + 8x + 1)

And that's the derivative of the given function f(x) = (x^2 + 3x - 2)/(2x + 1) using the quotient rule.