Differential calculus

In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve.

The primary objects of study in differential calculus are the derivative of a function, related notions such as the differential, and their applications. The derivative of a function at a chosen input value describes the rate of change of the function near that input value. Geometrically, the derivative at a point is the slope of the tangent line to the graph of the function at that point.

Differentiation has applications in nearly all quantitative disciplines. In physics, the derivative of the displacement of a moving body with respect to time is the velocity of the body, and the derivative of the velocity with respect to time is acceleration. Derivatives are also used to find the maxima and minima of a function, which helps solve many real-world optimization problems.

Derivative

Main article: Derivative

The derivative of a function tells us how steep a line is at any point on a graph. Think of it like this: if you have a hill, the derivative tells you how steep the hill is at a particular spot.

To understand this, start with straight lines. A straight line has the same steepness everywhere. We call this steepness the "slope." For example, if you move 3 steps up and 1 step right, the slope is 3 divided by 1, which is 3. But for curvy lines, the steepness changes depending on where you look. To find the steepness at one exact point, we use a line that just touches that point — called a tangent line. The slope of this tangent line is the derivative at that point.

We can find this by looking at points very close together on the curve and calculating the slope between them. As these points get closer, the slope gets closer to the true steepness at that single point. This idea helps us understand how things change at every moment — like how fast a car is going at an exact second, not just on average over a few seconds.

History of differentiation

Main article: History of calculus

The idea of a derivative, which helps us understand how things change, goes back a very long time. Ancient Greek mathematicians like Euclid and Archimedes thought about lines that just touch curves at one point. Much later, mathematicians such as Bhāskara II used very small values to study how things change.

The modern shape of calculus is mostly thanks to Isaac Newton and Gottfried Wilhelm Leibniz in the 1600s. They each came up with ways to describe change and showed how these ideas connect to adding up areas under curves. Many other mathematicians added to these ideas over time, making calculus the powerful tool we use today.

Applications of derivatives