Question:

Let f(x) = (2x^3 - x^2 + 3) / (4x^2 - x + 2) be a differentiable function. Determine f'(x) using the product and quotient rules.

Explanation:

To find the derivative of the function f(x) using the product and quotient rules, we will need to differentiate the numerator and denominator separately, and then apply the formulas for the product and quotient rules. Let's begin by finding the derivatives of the numerator and denominator:

Numerator (g(x)):

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

g'(x) = 6x^2 - 2x

Denominator (h(x)):

h(x) = 4x^2 - x + 2

h'(x) = 8x - 1

Now, let's apply the product rule to differentiate the function:

f'(x) = [(h(x) * g'(x)) - (g(x) * h'(x))] / [h(x)]^2

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

Expanding the expression, we get:

f'(x) = [24x^4 - 8x^2 - 6x^3 + 2x - 12x^2 + 4x - 48x^2 + 16x - 8 - 16x^4 + 8x^3 - 2x^2 + 64x^2 - 8x - 24] / [16x^4 - 8x^3 + 4x^2 + x^2 - 2x - 4x^2 + 2x - 8x + 1]

Combining like terms, we have:

f'(x) = [8x^4 - 14x^3 - 62x^2 + 14x - 32] / [16x^4 - 8x^3 + 2x^2 - 8x + 1]

Therefore, the derivative of f(x) is given by:

f'(x) = (8x^4 - 14x^3 - 62x^2 + 14x - 32) / (16x^4 - 8x^3 + 2x^2 - 8x + 1)