First and second derivatives Find f′(x),f′′(x).
f(x) = x/x+2

Verified step by step guidance

1. Step 1: Rewrite the function f(x) = \frac{x}{x+2} in a form suitable for differentiation. This can be done by recognizing it as a quotient of two functions, where the numerator u(x) = x and the denominator v(x) = x + 2.
2. Step 2: Apply the Quotient Rule for differentiation to find the first derivative f'(x). The Quotient Rule states that if you have a function h(x) = \frac{u(x)}{v(x)}, then h'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}.
3. Step 3: Differentiate the numerator and the denominator separately. For u(x) = x, the derivative u'(x) = 1. For v(x) = x + 2, the derivative v'(x) = 1.
4. Step 4: Substitute the derivatives u'(x) and v'(x) into the Quotient Rule formula to find f'(x). This gives f'(x) = \frac{1 \cdot (x+2) - x \cdot 1}{(x+2)^2}. Simplify the expression to get the first derivative.
5. Step 5: To find the second derivative f''(x), differentiate f'(x) with respect to x. This may involve using the Quotient Rule again if f'(x) is still in a quotient form, or using the Power Rule if it simplifies to a polynomial form.