The derivative of a function is the instantaneous rate of change of the function at a specific point. It represents the slope of the tangent line to the function at that point.
The derivative is defined as a limit. Specifically, it is the limit of the slope of a secant line as the two points on the line approach each other. This limit is taken as the distance between the two points approaches zero, resulting in the instantaneous rate of change or the derivative.
The derivative is an essential concept in calculus because it allows us to find the rate of change of a function at any point. This enables us to solve problems involving motion, optimization, and many other real-world applications.
The average rate of change is the average slope of a function over an interval, while the instantaneous rate of change is the slope of the tangent line to the function at a specific point. The average rate of change gives an overall picture of the function's behavior, while the instantaneous rate of change gives a more precise measurement at a specific point.
To find the derivative of a function using the definition, you need to take the limit of the slope of a secant line as the distance between two points on the line approaches zero. This limit will result in the instantaneous rate of change, which is the derivative of the function at that specific point.