Differential Calculus

At its core, differential calculus quantifies how a function responds to small changes in its input. This concept, formalized in the 17th century by Isaac Newton and Gottfried Wilhelm Leibniz, underpins physics, engineering, economics, machine learning, and countless modern technologies.

The central object of study is the derivative, which represents the instantaneous rate of change or the slope of the tangent line to a curve at a given point.

Limits & Continuity

Before defining the derivative, we must understand limits. A limit describes the value that a function approaches as the input approaches some value.

\lim_{x \to a} f(x) = L

Continuity requires that the limit exists, the function is defined at the point, and they match: \lim_{x \to a} f(x) = f(a). Differentiable functions are always continuous, but continuous functions are not necessarily differentiable (e.g., sharp corners).

Definition of the Derivative

Formal Definition

The derivative of a function f(x) at a point x = a is defined as:

f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}

This limit, if it exists, gives the instantaneous rate of change. Geometrically, it is the slope of the tangent line to the graph of f at (a, f(a)).

Alternative Notations

Depending on context, the derivative may be written as:

f'(x) — Lagrange's notation
\frac{dy}{dx} or \frac{df}{dx} — Leibniz's notation
\dot{y} — Newton's notation (common in physics for time derivatives)

Differentiation Rules

Computing derivatives from the limit definition is often impractical. Instead, we use established rules derived from the definition.

\frac{d}{dx}[c] = 0 \quad (Constant Rule)

\frac{d}{dx}[x^n] = nx^{n-1} \quad (Power Rule)

\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x) \quad (Sum/Difference)

\frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x) \quad (Constant Multiple)

\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x) \quad (Product Rule)

\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2} \quad (Quotient Rule)

\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x) \quad (Chain Rule)

The Chain Rule

The chain rule is arguably the most important tool in differential calculus, enabling differentiation of composite functions.

Why It Matters

Without the chain rule, differentiating functions like \sin(x^2) or e^{\ln(x)} would require unwieldy limit expansions. It decomposes complex functions into manageable layers.

Example: Differentiate y = (3x^2 + 2)^5

Let u = 3x^2 + 2, so y = u^5. Apply chain rule:

\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 5u^4 \cdot (6x) = 30x(3x^2 + 2)^4

Applications

Differential calculus extends far beyond abstract computation. It models real-world phenomena where change is continuous and measurable.

Optimization

By finding critical points where f'(x) = 0 or undefined, we can locate local maxima and minima. The second derivative test determines concavity:

f''(x) > 0 → Concave up (local minimum)
f''(x) < 0 → Concave down (local maximum)

Curve Sketching & Behavior

Derivatives reveal:

Intervals of increase/decrease
Local extrema
Concavity and inflection points
Asymptotic behavior (when combined with limits)

Related Rates & Physics

In kinematics, velocity is the derivative of position v(t) = s'(t), and acceleration is the derivative of velocity a(t) = v'(t). Related rates problems use implicit differentiation to find how one changing quantity affects another over time.

References & Further Reading

Stewart, J. Calculus: Early Transcendentals. 9th ed. Cengage Learning, 2020.
Apostol, T. M. Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra. Wiley, 2012.
Kurzweil, A. The 15 Mystery Solutions. MIT OpenCourseWare, 2023.
Aevum Encyclopedia Editorial Board. Foundations of Real Analysis. Retrieved Oct 2025.