Question:

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

Solution:

To find the derivative of a function using the product rule, we use the formula:

d/dx (f(x) * g(x)) = f'(x) * g(x) + f(x) * g'(x)

To find the derivative of a function using the quotient rule, we use the formula:

d/dx (f(x) / g(x)) = (f'(x) * g(x) - f(x) * g'(x)) / g(x)^2

Now, let's find the derivative of the given function using both the product rule and quotient rule:

Using the product rule: Let f(x) = 3x^2 - 2x and g(x) = x^3.

We have: f'(x) = 6x - 2 (derivative of f(x) with respect to x)*
g(x) = x^3 (original g(x) function)*
g'(x) = 3x^2 (derivative of g(x) with respect to x)*

Using the product rule formula, we can find the derivative of the function:

d/dx [(3x^2 - 2x) / x^3] = [(6x - 2) * x^3 - (3x^2)(3x^2 - 2x)] / x^6
Simplifying, we get:
[6x^4 - 2x^4 - 9x^4 + 6x^3] / x^6 = -5x^4 + 6x^3 / x^6 = (-5x^4 + 6x^3) / x^6

Using the quotient rule: Let f(x) = 3x^2 - 2x and g(x) = x^3.

We have: f'(x) = 6x - 2 (derivative of f(x) with respect to x)*
g(x) = x^3 (original g(x) function)*
g'(x) = 3x^2 (derivative of g(x) with respect to x)*

Using the quotient rule formula, we can find the derivative of the function:

d/dx [(3x^2 - 2x) / x^3] = [(6x - 2)(x^3) - (3x^2)(3x^2 - 2x)] / (x^3)^2
Simplifying, we get:
(6x^4 - 2x^3 - 9x^4 + 6x^3) / x^6 = (-3x^4 + 4x^3) / x^6

Therefore, the derivative of the function f(x) = (3x^2 - 2x) / x^3 using both the product rule and quotient rule is:

-5x^4 + 6x^3 / x^6 or (-3x^4 + 4x^3) / x^6

Note: Either form is correct, and you can choose to simplify it further if necessary.