Lecture 20: Differentials of Functions

1.1 Definition of a Differential

1.1.1 Why Differentials Are Useful

Core Idea

• A differential is the linear part of a small change in a function. It approximates the actual increment when the independent-variable increment is small.
• A derivative describes an instantaneous rate of change; a differential uses that rate to approximate an actual small change.

Example

• Let $f(x)=x^2$. At $x_0$,

• As $\Delta x\to0$, the term $(\Delta x)^2$ is of higher order than $\Delta x$, so

$$\Delta y\approx2x_0\Delta x.$$

• The linear term is the differential:

$$dy=f'(x_0)\,dx=2x_0\Delta x,\qquad dx=\Delta x.$$

• This approximation is useful in error estimation.
• 

Relationship Between a Derivative and a Differential

• The derivative $f'(x_0)$ is the local rate of change.
• The actual increment is $\Delta y=f(x_0+\Delta x)-f(x_0)$.
• The differential $dy=f'(x_0)\Delta x$ is the linear principal part of $\Delta y$.
• Therefore,

$$\frac{dy}{dx}=f'(x_0).$$

1.1.2 Formal Definition

#####Definition: Differential

Description: If

$$f(x_0+\Delta x)-f(x_0)=A\Delta x+o(\Delta x)\qquad(\Delta x\to0),$$

then $f$ is differentiable at $x_0$. Its differential is

$$dy=A\Delta x.$$

Explanation

• The term $A\Delta x$ is the linear principal part of $\Delta y$.
• The remainder $o(\Delta x)$ is a higher-order infinitesimal.
• Hence

$$\Delta y=dy+o(\Delta x),\qquad dy\approx\Delta y.$$

Geometric Meaning

• For the curve $y=f(x)$, $dy=f'(x_0)dx$ is the vertical increment along the tangent line at $x_0$ corresponding to a horizontal increment $dx$.

1.1.3 Differentiability and Existence of the Derivative

#####Theorem: DifferentiabilityAndDerivative

Description: For a single-variable function, $f$ is differentiable at $x_0$ if and only if $f'(x_0)$ exists. In that case,

$$dy=f'(x_0)\,dx.$$

1.2 Continuity, Differentiability, and the Derivative

Illustration

Main Relationships

• Differentiability at $x_0$ implies continuity at $x_0$.
• Continuity does not imply differentiability. For example, $f(x)=|x|$ is continuous at $0$, but its one-sided derivatives are $-1$ and $1$.
• Existence of $f'(x_0)$ does not imply that $f'$ is continuous at $x_0$.
• Existence of $f''(x_0)$ does not by itself imply that $\lim_{x\to x_0}f''(x)$ exists.

Proof That Differentiability Implies Continuity

• If $f'(x_0)$ exists, then

$$\Delta y=\frac{\Delta y}{\Delta x}\Delta x.$$

• As $\Delta x\to0$, the difference quotient tends to the finite number $f'(x_0)$, while $\Delta x\to0$. Therefore $\Delta y\to0$, which is continuity at $x_0$.

How to Test Differentiability

• If the function is discontinuous at the point, it is not differentiable there.
• If it is continuous, differentiability still must be checked:
• use the derivative definition; or
• verify that the left- and right-hand derivatives both exist and are equal.

1.2.2 Repeated Use of L'Hôpital's Rule

1. Reapply the rule only while the expression remains of type $0/0$ or $\infty/\infty$ and all hypotheses of the rule continue to hold.
2. At every application, check that the derivative of the denominator is nonzero in a punctured neighborhood and that the new derivative ratio has a limit.

1.3 Supplement: Historical View of Differentials

1.3.1 Classical Calculus

Characteristics

1. Early calculus treated $dx$ and $dy$ as infinitesimal changes.
2. Tangents were described using these infinitesimals.
3. The derivative was interpreted as the tangent slope $dy/dx$.

1.3.2 Limit-Based Definition

Limits provide a precise meaning for infinitesimals: a quantity is infinitesimal when its limit is zero.

Modern Interpretation

• In general, $dy$ is not identical to $\Delta y$.
• Instead,

$$\Delta y=dy+o(dx),\qquad dy\approx\Delta y\quad(dx\to0).$$

• Thus the differential is the best linear approximation to the actual increment.