Differential calculus

Differential calculus is one of the most fundamental disciplines in analysis. It deals with how a quantity changes and is used to describe and predict variations in physical, biological, and economic systems.

 

A central idea in differential calculus is to determine a function’s instantaneous rate of change — that is, how fast its value changes at a specific point. This is expressed through the derivative function, which describes the slope of the tangent line that touches the graph at that point.

 

 

Connection with limits and continuity

To talk about the rate of change, one must examine how a function behaves as the input approaches a certain value. Here, the concept of a limit is used. A function must also be continuous to be differentiable, which means its graph can be drawn without breaks or jumps.

 

$$ \large f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} $$

 

This formula shows that differential calculus is a natural extension of the limit concept: one calculates the slope as the limit of the secant’s slope as two points on the graph approach each other.

 

Differentiation

To differentiate means to determine the derivative function \( \large f'(x) \), which describes how fast \( \large f(x) \) changes with respect to \( \large x \). If \( \large f(x) \) represents a distance, \( \large f'(x) \) represents the speed; if \( \large f(x) \) represents a temperature, \( \large f'(x) \) shows how the temperature changes over time.

 

 

Historical context

Differential calculus was developed in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz. Both independently formulated the principles of what we now call calculus. Although their notation was different, they described the same phenomenon: the relationship between motion, speed, and change.

 

 

Intuition: secant, tangent, and rate of change

Imagine the graph of a function \( \large f(x) \). If you connect two points on the graph with a straight line, you get a secant. The slope of the secant indicates the average rate of change between the two points. As the points move closer together, the secant approaches a line that touches the graph at only one point — the tangent.

 

The slope of this tangent at a point is called the instantaneous rate of change and is exactly the value that differential calculus seeks to compute. It describes how the function changes precisely at that point.