Differential calculus

In mathematics, differential calculus is a part of calculus that looks at how things change. It is one of two main parts of calculus. The other part is integral calculus, which looks at the area under a curve.

The main idea in differential calculus is the derivative of a function. The derivative tells us how fast a function is changing at a certain point. If we think about a graph, the derivative at a point is the slope of a line that just touches the graph at that point, called the tangent line.

We can use derivatives in many areas. In physics, the derivative of how far something has moved (displacement) over time tells us the velocity of that object. The derivative of velocity over time tells us the acceleration. Derivatives also help us find the highest and lowest points of a function, which can solve many practical problems.

Derivative

Main article: Derivative

The derivative of a function tells us how steep a line is at any point on a graph. Imagine you are on a hill. The derivative tells you how steep the hill is at that exact 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 helps us understand how things change at every moment — like how fast a car is going at one exact second.

History of differentiation

Main article: History of calculus

The idea of a derivative helps us understand how things change. It has been around for a very long time. Ancient Greek mathematicians like Euclid and Archimedes thought about lines that touch curves at just one point.

Later, mathematicians such as Bhāskara II used very small values to study change.

The modern shape of calculus was mostly developed by Isaac Newton and Gottfried Wilhelm Leibniz in the 1600s. They found ways to describe change and showed how these ideas connect to areas under curves. Many other mathematicians added to these ideas, making calculus the useful tool we have today.

Applications of derivatives