21–30. Derivatives
a. Use limits to find the derivative function f' for the following functions f.
f(x) = 1/x+1; a = -1/2;5

Verified step by step guidance

1. Step 1: Understand the problem. We need to find the derivative of the function f(x) = \frac{1}{x+1} using the definition of the derivative, which involves limits.
2. Step 2: Recall the definition of the derivative. The derivative f'(x) of a function f(x) at a point x is given by the limit: f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}.
3. Step 3: Substitute f(x) = \frac{1}{x+1} into the definition. This gives us: f'(x) = \lim_{h \to 0} \frac{\frac{1}{x+h+1} - \frac{1}{x+1}}{h}.
4. Step 4: Simplify the expression inside the limit. Find a common denominator for the fractions in the numerator: \frac{1}{x+h+1} - \frac{1}{x+1} = \frac{(x+1) - (x+h+1)}{(x+h+1)(x+1)} = \frac{-h}{(x+h+1)(x+1)}.
5. Step 5: Substitute the simplified expression back into the limit and simplify further: f'(x) = \lim_{h \to 0} \frac{-h}{h(x+h+1)(x+1)}. Cancel the h in the numerator and denominator, then evaluate the limit as h approaches 0.

Key Concepts

Derivatives

A derivative represents the rate of change of a function with respect to its variable. It is defined as the limit of the average rate of change of the function as the interval approaches zero. In calculus, the derivative is often denoted as f'(x) and can be interpreted as the slope of the tangent line to the graph of the function at a given point.

Limits

Limits are a fundamental concept in calculus that describe the behavior of a function as its input approaches a certain value. They are essential for defining derivatives, as the derivative is calculated using the limit of the difference quotient. Understanding limits allows us to analyze functions at points where they may not be explicitly defined or where they exhibit discontinuities.

Difference Quotient

The difference quotient is a formula used to calculate the average rate of change of a function over an interval. It is expressed as (f(x+h) - f(x))/h, where h is the change in x. As h approaches zero, the difference quotient approaches the derivative of the function, providing a way to find instantaneous rates of change.