Answer:
The derivative of f(x) = 3 ÷ x + 2 is f'(x) = -3 ÷ (x+2)².
Explanation:
This is found by using the first principle method of differentiation, which states that the derivative of a function is the rate of change of the function with respect to the indepedent variable. In other words, the derivative of a function measures how much the output of the function changes with respect to a small change in the input.
To find the derivative of f(x) = 3 ÷ x + 2, we start by taking the limit as h approaches 0 of the difference quotient (f(x + h) - f(x)) ÷ h.
The difference quotient is (3 ÷ (x + h + 2) - 3 ÷ (x + 2)) ÷ h. Simplified, this becomes 3h ÷ (x + h + 2)(x + 2).
When h approaches 0, the difference quotient becomes 3 ÷ (x + 2)², which is the derivative of f(x).