📈

Derivative: Calculus Study Notes

October 10, 2026

📈 Mathematical Derivatives: A Comprehensive Guide

  • Overview of the derivative concept and its geometric meaning
  • Formal definitions using limits and infinitesimals
  • Relationship between continuity and differentiability
  • Standard notations across different mathematical contexts
  • Fundamental computation rules and formulas for basic and combined functions

💡 Core Concepts and Intuition

In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input.

  • Geometric Interpretation: The derivative of a single-variable function at a chosen input value is the slope of the tangent line to the graph of the function at that specific point.
  • Linear Approximation: The tangent line represents the best linear approximation of the function near that input value.
  • Rate of Change: It is frequently described as the instantaneous rate of change—the ratio of the instantaneous change in the dependent variable to that of the independent variable.
  • Differentiation: The process of finding a derivative is known as differentiation.

Multivariable Generalization

Derivatives can be generalized to functions of several real variables:

  • Linear Transformation: Reinterpreted as a linear transformation whose graph provides the best linear approximation to the original function's graph.
  • Jacobian Matrix: A matrix representing this linear transformation with respect to the chosen basis of independent and dependent variables, calculated using partial derivatives.
  • Gradient Vector: For a real-valued function of several variables, the Jacobian matrix simplifies to the gradient vector.

📐 Formal Definitions of the Derivative

1. Definition as a Limit

A function f(x)f(x) of a real variable is differentiable at a point aa if its domain contains an open interval containing aa, and the following limit exists:

L=lim⁡h→0f(a+h)−f(a)hL = \lim_{h\to 0}{\frac{f(a+h)-f(a)}{h}}

The (ε,δ)(\varepsilon, \delta)-Definition

For every positive real number ε\varepsilon, there exists a positive real number δ\delta such that for every hh satisfying ∣h∣<δ|h| < \delta and h≠0h \neq 0:

  • f(a+h)f(a+h) is defined
  • ∣L−f(a+h)−f(a)h∣<ε\left|L - \frac{f(a+h)-f(a)}{h}\right| < \varepsilon (where vertical bars denote absolute value)

The Derivative Function

  • If the limit LL exists at aa, it is called the derivative of ff at aa, denoted f′(a)f'(a) or dfdx(a)\frac{df}{dx}(a).
  • If ff has a derivative at every point in its domain, the mapping x↦f′(x)x \mapsto f'(x) defines the derivative function f′f'.
  • The domain of f′f' may be smaller than the domain of ff if f′f' is undefined at certain points.

Example: The Squaring Function

Let f(x)=x2f(x) = x^2. The difference quotient is:

f(a+h)−f(a)h=(a+h)2−a2h=a2+2ah+h2−a2h=2a+h\frac{f(a+h)-f(a)}{h} = \frac{(a+h)^2-a^2}{h} = \frac{a^2+2ah+h^2-a^2}{h} = 2a+h

As hh approaches 00, the expression approaches 2a2a. Thus, the derivative of the squaring function is the doubling function: f′(x)=2xf'(x) = 2x.

2. Definition Using Infinitesimals

The derivative dfdx(a)\frac{df}{dx}(a) can be viewed as the ratio of an infinitesimal change in output to an infinitesimal change in input.

  • Hyperreal Numbers: Nonstandard analysis uses hyperreals—an extension of real numbers containing infinite numbers whose reciprocals are infinitesimals—to make this intuition rigorous.
  • Standard Part Function (st⁡\operatorname{st}): Rounds off finite hyperreals to the nearest real number.
  • Leibniz Notation Formulation:

f′(x)=st⁡(f(x+dx)−f(x)dx)f'(x) = \operatorname{st} \left(\frac{f(x+dx)-f(x)}{dx}\right)

Example with f(x)=x2f(x) = x^2:

f'(x) &= \operatorname{st} \left(\frac{x^2+2x\cdot dx+(dx)^2-x^2}{dx}\right)\\ &= \operatorname{st} \left(\frac{2x\cdot dx+(dx)^2}{dx}\right)\\ &= \operatorname{st} \left(\frac{2x\cdot dx}{dx}+\frac{(dx)^2}{dx}\right)\\ &= \operatorname{st} \left(2x+dx\right)\\ &= 2x. \end{aligned}$$ --- ## 🔄 Continuity and Differentiability - **Differentiability Implies Continuity**: If $f$ is differentiable at $a$, it **must be continuous** at $a$. - **Continuous Does Not Imply Differentiable**: There are continuous functions that lack derivatives entirely or at specific points. ### Common Exceptions to Differentiability - **Step Functions**: Secant lines approach infinity; limits do not exist. - **Kinks and Cusps**: For example, $f(x) = |x|$ is continuous at $x = 0$ but has a "kink," yielding a slope of $1$ from the right and $-1$ from the left. - **Vertical Tangents**: Smooth graphs with vertical tangents lack derivatives (e.g., $f(x) = x^{1/3}$ at $x = 0$). - **Weierstrass Function**: Discovered in 1872, it is continuous everywhere but differentiable nowhere. - **Meager Sets**: Stefan Banach proved in 1931 that functions having derivatives at some points form a meager set among continuous functions, meaning random continuous functions rarely have derivatives. --- ## ✍️ Notations for Differentiation | Notation Name | Symbol / Representation | Key Characteristics & Usage | |:--------------|:-------------------------|:---------------------------| | **Leibniz Notation** | $\frac{dy}{dx}$, $\frac{d^n y}{dx^n}$ | - Ratio of differentials introduced by Gottfried Wilhelm Leibniz (1675).<br/>- Explicitly specifies the variable in the denominator.<br/>- Ideal for chain rule and multiple interrelated quantities. | | **Prime Notation** | $f'(x)$, $y'$, $f^{(n)}$ | - Uses prime marks, introduced by Joseph-Louis Lagrange.<br/>- Higher orders use multiple primes, Roman numerals ($f^{\mathrm{iv}}$), or parentheses ($f^{(4)}$). | | **Newton's (Dot) Notation** | $\dot{y}$, $\ddot{y}$ | - Uses dots placed over symbols.<br/>- **Used exclusively for time or arc length derivatives** in physics and differential geometry.<br/>- Unmanageable for orders $\ge 4$ or multiple variables. | | **Euler (D) Notation** | $Df(x)$, $D^n f(x)$, $D_x u$ | - Treats $D$ as a differential operator (introduced by Louis François Antoine Arbogast).<br/>- Partial derivatives indicated via subscripts ($D_{xy}f$). | --- ## ⚙️ Rules of Computation Differentiation rules allow us to compute derivatives of complex functions by combining simple base functions. ### 1. Rules for Basic Functions | Function Type | Function $f(x)$ | Derivative $f'(x)$ | Conditions / Notes | |:--------------|:----------------|:-------------------|:-------------------| | **Powers** | $x^a$ | $ax^{a-1}$ | $a$ is a real number | | **Exponential** | $e^x$ | $e^x$ | $e \approx 2.71828$ | | **Exponential (General)** | $a^x$ | $a^x \ln(a)$ | $a > 0$ | | **Natural Logarithm** | $\ln(x)$ | $\frac{1}{x}$ | $x > 0$ | | **Logarithm (General)** | $\log_a(x)$ | $\frac{1}{x\ln(a)}$ | $x, a > 0$ | | **Sine** | $\sin(x)$ | $\cos(x)$ | - | | **Cosine** | $\cos(x)$ | $-\sin(x)$ | - | | **Tangent** | $\tan(x)$ | $\sec^2(x) = \frac{1}{\cos^2(x)} = 1+\tan^2(x)$ | - | | **Arcsine** | $\arcsin(x)$ | $\frac{1}{\sqrt{1-x^2}}$ | $-1 < x < 1$ | | **Arccosine** | $\arccos(x)$ | $-\frac{1}{\sqrt{1-x^2}}$ | $-1 < x < 1$ | | **Arctangent** | $\arctan(x)$ | $\frac{1}{1+x^2}$ | - | --- ### 2. Rules for Combined Functions - **Constant Rule**: If $f$ is a constant function, then $f'(x) = 0$. - **Sum Rule**: $$(\alpha f + \beta g)' = \alpha f' + \beta g'$$ for all functions $f, g$ and real numbers $\alpha, \beta$. - **Product Rule**: $$(fg)' = f'g + fg'$$ *Special Case*: $(\alpha f)' = \alpha f'$ for constant $\alpha$. - **Quotient Rule**: $$\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$$ at all inputs where $g \neq 0$. - **Chain Rule (Composite Functions)**: If $f(x) = h(g(x))$, then: $$f'(x) = h'(g(x)) \cdot g'(x)$$