Applications of derivatives
A large part of early calculus involves learning how to differentiate various types of functions using concepts such as the quotient and product rules, the chain rule, implicit differentiation, and more. The reason that a significant portion of time is spent learning how to compute derivatives is because derivatives have many important applications including:
- Determining the rate of change of some quantity
- Determining many characteristics of the graph of a function
- Optimization
These are just a few examples, there are many other applications of derivatives.
Rate of change
Using derivatives to determine the rate of change of some quantity is a common application of derivatives. Given a function, f(x), the derivative of the function, f'(x), represents the rate of change of the function f(x). Geometrically, f'(x) is the slope of the line tangent to the curve of f(x) at a given point. Thus, the derivative of a function at a given point is the rate of change of the function at that point. This fact can be used to solve numerous types of real-world problems as well as determine characteristics of a function such as the intervals over which a function is increasing, decreasing, or not changing:
- If f'(x) > 0 for all x over an interval, f(x) is increasing over the interval.
- If f'(x) < 0 for all x over an interval, f(x) is decreasing over the interval.
- If f'(x) = 0, f(x) is not changing.
Characteristics of a graph
Many characteristics of the graph of f(x) can be determined using derivatives, such as whether the function is increasing or decreasing, the location(s) of its local extrema and absolute extrema, its critical points, inflection points, concavity, and more.
Being able to determine these various characteristics allows us to graph functions without needing tools such as graphing calculators, which can be useful in certain cases.
Optimization
Optimization problems involve finding absolute extrema of functions that model some problem we want to optimize. Both the first derivative and second derivative of a function can be used to find absolute extrema.
Optimization problems have many real world applications. As long as whatever is being studied can be modeled by a function, derivatives can be used to optimize the function, even allowing us to optimize functions given certain constraints.