Derivatives Find and simplify the derivative of the following functions.
f(x) = x /x+1

Verified step by step guidance

1. Step 1: Recognize that the function \( f(x) = \frac{x}{x+1} \) is a quotient of two functions, \( u(x) = x \) and \( v(x) = x+1 \). To find the derivative, we will use the Quotient Rule.
2. Step 2: Recall the Quotient Rule, which states that if \( f(x) = \frac{u(x)}{v(x)} \), then \( f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} \).
3. Step 3: Differentiate \( u(x) = x \) to get \( u'(x) = 1 \), and differentiate \( v(x) = x+1 \) to get \( v'(x) = 1 \).
4. Step 4: Substitute \( u(x) \), \( u'(x) \), \( v(x) \), and \( v'(x) \) into the Quotient Rule formula: \( f'(x) = \frac{1 \cdot (x+1) - x \cdot 1}{(x+1)^2} \).
5. Step 5: Simplify the expression obtained in Step 4 to find the simplified form of the derivative.