21–30. Derivatives
b. Evaluate f'(a) for the given values of a.
f(x) = 1/x+1; a = -1/2;5

Verified step by step guidance

1. Step 1: Identify the function f(x) = \frac{1}{x} + 1 and the points a = -\frac{1}{2} and a = 5 where we need to evaluate the derivative f'(a).
2. Step 2: Find the derivative of the function f(x). The function can be rewritten as f(x) = x^{-1} + 1. Use the power rule for derivatives, which states that \frac{d}{dx}[x^n] = nx^{n-1}, to find f'(x).
3. Step 3: Apply the power rule to f(x) = x^{-1} + 1. The derivative of x^{-1} is -x^{-2}, and the derivative of a constant (1) is 0. Therefore, f'(x) = -x^{-2}.
4. Step 4: Evaluate f'(x) at a = -\frac{1}{2}. Substitute x = -\frac{1}{2} into f'(x) = -x^{-2} to find f'(-\frac{1}{2}).
5. Step 5: Evaluate f'(x) at a = 5. Substitute x = 5 into f'(x) = -x^{-2} to find f'(5).

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 a given point. The derivative can be computed using various rules, such as the power rule, product rule, and quotient rule, depending on the form of the function.

Quotient Rule

The quotient rule is a method for finding the derivative of a function that is the ratio of two other functions. If f(x) = g(x)/h(x), the derivative f'(x) is given by f'(x) = (g'(x)h(x) - g(x)h'(x)) / (h(x))^2. This rule is essential when dealing with functions that involve division, such as the function f(x) = 1/(x+1) in the given question.

Evaluating Derivatives at Specific Points

Evaluating the derivative at a specific point involves substituting the value of that point into the derivative function. This process provides the slope of the tangent line to the function at that particular point, which can be useful for understanding the behavior of the function. In this case, we need to compute f'(-1/2) and f'(5) to analyze the function's behavior at these values.