Derivatives are a fundamental concept in calculus that represent the rate of change of a function at a given point. They are used to analyze the behavior of functions, find maximum and minimum values, and understand the relationship between different variables. The process of finding a derivative is called differentiation.
In simpler terms, a derivative measures how a function changes as its input (usually denoted as $$x$$) changes. It can be thought of as the slope of a tangent line to the graph of the function at a specific point. The derivative of a function $$f(x)$$ is denoted as $$f(x)$$ or $$\frac{df}{dx}$$. The derivative of a function represents the rate at which the function is changing at a given point.
For example, the derivative of the function $$f(x) = x^2$$ is $$f'(x) = 2x$$. This means that the slope of the tangent line to the graph of $$f(x) = x^2$$ at any point $$x$$ is equal to $$2x$$. So, when $$x = 2$$, the slope of the tangent line is $$2(2) = 4$$, and when $$x = 5$$, the slope of the tangent line is $$2(5) = 10$$.
The derivative of a function can also be thought of as the function that gives the slope of the original function at each point. For example, the derivative of $$f(x) = x^2$$ is $$f'(x) = 2x$$. This means that the slope of the tangent line to the graph of $$f(x) = x^2$$ at any point $$x$$ is given by the function $$2x$$.
Derivatives have many applications in various fields, including physics, economics, and engineering. They can be used to find the maximum and minimum values of functions, determine the concavity of a graph, and solve optimization problems.
