find the derivative of the function
(x^3-3x^2+4)/x^2
To find the derivative of the function, we can use the quotient rule. The quotient rule states that for a function of the form f(x) = g(x)/h(x), where g(x) and h(x) are differentiable functions, the derivative f'(x) can be calculated as:
f'(x) = (g'(x)*h(x) - g(x)*h'(x)) / (h(x))^2
In this case, g(x) = x^3 - 3x^2 + 4 and h(x) = x^2. Let's find the derivative step by step.
Step 1: Find g'(x)
The derivative of g(x) can be found by applying the power rule. The power rule states that the derivative of x^n is n*x^(n-1).
g'(x) = 3x^2 - 6x
Step 2: Find h'(x)
The derivative of h(x) is also found by applying the power rule.
h'(x) = 2x
Step 3: Apply the quotient rule
Now, substitute g'(x), h(x), and h'(x) into the quotient rule formula.
f'(x) = ((3x^2 - 6x)*x^2 - (x^3 - 3x^2 + 4)*2x) / (x^2)^2
Simplifying this expression will give you the derivative of the original function.