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

Verified step by step guidance

1. Step 1: Recall the definition of the derivative using limits. The derivative of a function f at a point a is given by the limit: f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}.
2. Step 2: Substitute the given function f(x) = 5x + 2 into the derivative definition. This means you need to find f(a+h) and f(a).
3. Step 3: Calculate f(a+h) by substituting x = a + h into the function: f(a+h) = 5(a+h) + 2.
4. Step 4: Calculate f(a) by substituting x = a into the function: f(a) = 5a + 2.
5. Step 5: Substitute f(a+h) and f(a) into the limit definition: f'(a) = \lim_{h \to 0} \frac{5(a+h) + 2 - (5a + 2)}{h}. Simplify the expression inside the limit.

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 provides critical information about the function's behavior, such as its slope at any given point.

Limits

Limits are fundamental to calculus and describe the behavior of a function as it approaches a particular point. They are used to define derivatives, as the derivative is essentially the limit of the difference quotient as the interval approaches zero. Understanding limits is crucial for evaluating the continuity and differentiability of functions.

Difference Quotient

The difference quotient is a formula that expresses the average rate of change of a function over an interval. It is given by (f(x+h) - f(x))/h, where h is the change in x. As h approaches zero, the difference quotient approaches the derivative, providing a way to calculate the instantaneous rate of change of the function at a specific point.