2.2 Defining the Derivative of a Function and Using Derivative Notation

In this guide, we will be exploring the concept of derivatives in Calculus AB. A derivative is a measure of how a function is changing at a specific point. It is also known as the instantaneous rate of change of the function. The derivative of a function at a point x is represented by the notation f'(x). Understanding the Definition of a Derivative: The derivative of a function can be represented using various forms of notation, such as f'(x), dy/dx, or the limit definition. The limit definition states that the derivative of a function f(x) at a point x is the limit as h approaches zero of (f(x+h) - f(x))/h. This concept can be difficult to grasp, but it is important to understand that the derivative is measuring the rate of change at a specific point, rather than over a range of values. Exploring the Derivative through Different Methods: One way to understand the derivative is to look at it graphically. The derivative of a function at a point can be represented by the slope of the line tangent to the graph of the function at that point. By observing the slope of the tangent line, we can see how steep the function is at that point and understand the rate of change. Another way to explore the derivative is numerically. This can be done by taking the difference quotient of a function and seeing how it changes as the interval becomes smaller and smaller. This method can give us an idea of the rate of change at a specific point by looking at how the function changes over a small interval. We can also explore the derivative analytically. This can be done by finding the equation of the line tangent to the graph of the function at a specific point. By finding the equation of the tangent line, we can see the slope of the line and understand the rate of change at that point. Lastly, we can explore the derivative verbally. This can be done by describing the rate of change of a function in words. For example, if a function represents the position of an object over time, we can describe the derivative as the velocity of the object at a specific point in time. Example Problems: Example 1: Consider the function f(x) = x^2. At the point x = 2, what is the rate of change of the function? Solution: To find the rate of change at the point x = 2, we can use the limit definition of the derivative. We can see that as the interval h approaches zero, the difference quotient (f(x+h) - f(x))/h approaches 4. Therefore, the derivative of the function f(x) = x^2 at the point x = 2 is 4. Example 2: Consider the function g(x) = 3x^2 + 2x. At the point x = 1, what is the rate of change of the function? Solution: To find the rate of change at the point x = 1, we can use the limit definition of the derivative. We can see that as the interval h approaches zero, the difference quotient (g(x+h) - g(x))/h approaches 5. Therefore, the derivative of the function g(x) = 3x^2 + 2x at the point x = 1 is 5. Example 3: Consider the function h(x) = (x^3 + 2x)/(x+1). At the point x = 2, what is the rate of change of the function?

Solution:

To find the rate of change at the point x = 2, we can use the limit definition of the derivative. We can see that as the interval h approaches zero, the difference quotient (h(x+h) - h(x))/h approaches 8. Therefore, the derivative of the function h(x) = (x^3 + 2x)/(x+1) at the point x = 2 is 8.

In conclusion, derivatives are a fundamental concept in Calculus AB that allow us to understand the rate of change of a function at a specific point. By exploring the derivative through various methods such as graphically, numerically, analytically and verbally, we can gain a better understanding of this important concept. The examples provided in this guide are just a small sample of the types of problems you may encounter when working with derivatives. As you continue to study Calculus AB, it will be important to practice solving various types of derivative problems to solidify your understanding of this topic.