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.

Key Concepts

Derivatives

A derivative represents the rate of change of a function with respect to its variable. It is a fundamental concept in calculus that provides information about the slope of the tangent line to the curve of the function at any given point. The derivative can be computed using various rules, such as the power rule, product rule, quotient rule, and chain rule.

Quotient Rule

The quotient rule is a method for finding the derivative of a function that is the ratio of two other functions. If you have a function f(x) = g(x)/h(x), the derivative f'(x) is given by (g'(x)h(x) - g(x)h'(x)) / (h(x))^2. This rule is essential when differentiating functions that are expressed as fractions.

Simplification of Derivatives

After finding the derivative of a function, simplification is often necessary to express the result in its simplest form. This may involve factoring, reducing fractions, or combining like terms. Simplifying the derivative can make it easier to analyze the function's behavior, such as identifying critical points and determining concavity.