A derivative is one of the most important concepts in calculus. It measures how fast a function changes at any given point. Think of it as the instantaneous rate of change. Geometrically, the derivative at a point gives us the slope of the tangent line to the curve at that exact point.
The derivative has a precise mathematical definition using limits. We define f prime of x as the limit as h approaches zero of the difference quotient: f of x plus h minus f of x, all divided by h. This formula captures the idea of finding the slope of a secant line as the distance h gets smaller and smaller, eventually becoming the tangent line.
There are several basic rules for finding derivatives that make calculations much easier. The power rule states that the derivative of x to the n is n times x to the n minus one. The derivative of any constant is zero. And the derivative of a sum is the sum of the derivatives. For example, the derivative of x squared is two x, which we can see graphically where the parabola x squared has derivative two x, a straight line.
Derivatives have countless real-world applications. In physics, the derivative of position with respect to time gives us velocity, and the derivative of velocity gives us acceleration. In economics, derivatives help us find marginal costs and revenues. In optimization problems, we use derivatives to find maximum and minimum values. Here we see a position function and its derivative, the velocity, with a moving point showing how velocity represents the rate of change of position.
To summarize what we have learned about derivatives: They measure the instantaneous rate of change of functions. Geometrically, they give us the slope of tangent lines. They are formally defined using limits of difference quotients. Basic derivative rules make calculations much easier. And derivatives are essential tools in physics, economics, optimization, and many other fields.