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. The process of finding a derivative is called differentiation. Geometrically, the derivative at a point is the slope of the tangent line to the graph of the function at that point, provided that the derivative exists and is defined at that point. For a real-valued function of a single real variable, the derivative of a function at a point generally determines the best linear approximation to the function at that point.

Differential calculus and integral calculus are connected by the fundamental theorem of calculus. This states that differentiation is the reverse process to integration.

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. The derivative of the momentum of a body with respect to time equals the force applied to the body; rearranging this derivative statement leads to the famous F = ma equation associated with Newton's second law of motion. The reaction rate of a chemical reaction is a derivative. In operations research, derivatives determine the most efficient ways to transport materials and design factories.

Derivatives are frequently used to find the maxima and minima of a function. Equations involving derivatives are called differential equations and are fundamental in describing natural phenomena. Derivatives and their generalizations appear in many fields of mathematics, such as complex analysis, functional analysis, differential geometry, measure theory, and abstract algebra.

The derivative of ${\displaystyle f(x)}$ at the point ${\displaystyle x=a}$ is the slope of the tangent to ${\displaystyle (a,f(a))}$. In order to gain an intuition for this, one must first be familiar with finding the slope of a linear equation, written in the form ${\displaystyle y=mx+b}$. The slope of an equation is its steepness. It can be found by picking any two points and dividing the change in ${\displaystyle y}$ by the change in ${\displaystyle x}$, meaning that ${\displaystyle {\text{slope }}={\frac {{\text{ change in }}y}{{\text{change in }}x}}}$. For, the graph of ${\displaystyle y=-2x+13}$ has a slope of ${\displaystyle -2}$, as shown in the diagram below:

For brevity, ${\displaystyle {\frac {{\text{change in }}y}{{\text{change in }}x}}}$ is often written as ${\displaystyle {\frac {\Delta y}{\Delta x}}}$, with ${\displaystyle \Delta }$ being the Greek letter delta, meaning 'change in'. The slope of a linear equation is constant, meaning that the steepness is the same everywhere. However, many graphs such as ${\displaystyle y=x^{2}}$ vary in their steepness. This means that you can no longer pick any two arbitrary points and compute the slope. Instead, the slope of the graph can be computed by considering the tangent line—a line that 'just touches' a particular point. The slope of a curve at a particular point is equal to the slope of the tangent to that point. For example, ${\displaystyle y=x^{2}}$ has a slope of ${\displaystyle 4}$ at ${\displaystyle x=2}$ because the slope of the tangent line to that point is equal to ${\displaystyle 4}$:

The derivative of a function is then simply the slope of this tangent line. Even though the tangent line only touches a single point at the point of tangency, it can be approximated by a line that goes through two points. This is known as a secant line. If the two points that the secant line goes through are close together, then the secant line closely resembles the tangent line, and, as a result, its slope is also very similar: