Explanation:
To find the first derivative of the function F(x) = (x - 3) / (x^2), we can use the quotient rule. The quotient rule states that if we have a function in the form f(x) = g(x) / h(x), where g(x) and h(x) are both differentiable functions, then the derivative of f(x) can be calculated as:
f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / (h(x))^2
In our case, g(x) = (x - 3) and h(x) = x^2. Let's calculate the first derivative:
F'(x) = [(1) * (x^2) - (x - 3) * (2x)] / (x^2)^2
= (x^2 - 2x(x - 3)) / x^4
= (x^2 - 2x^2 + 6x) / x^4
= (-x^2 + 6x) / x^4
= -x(x - 6) / x^4
= -(x - 6) / x^3
Therefore, the first derivative of F(x) = (x - 3) / (x^2) is F'(x) = -(x - 6) / x^3.