A derivative is a mathematical tool used to measure the rate of change of a function at a specific point. It is calculated as the slope of a tangent line to the curve of the function at that point.
Derivatives are used in a wide range of fields, including physics, economics, engineering, and more. Some of the main applications include optimization, finding maximum and minimum values, and determining rates of change.
The chain rule is a method for finding the derivative of a composite function, which is a function that is made up of two or more other functions. It is a crucial tool in the application of derivatives as it allows us to find the derivative of more complex functions.
A local extremum is a point on a curve where the function reaches a maximum or minimum value, but only in a certain interval or region. A global extremum is a point where the function reaches the maximum or minimum value over its entire domain.
Derivatives can give us information about the shape of a graph, such as whether the curve is increasing or decreasing, or whether it has a concave up or concave down shape. They can also help us identify critical points, where the slope of the tangent line is zero, and determine whether they are maximum or minimum points.