nLab Newton-Leibniz operator

Definition

Given an Archimedean field $F$, an open interval $I \subseteq F$, a pointwise continuous function $f:C^0(I, F)$ is differentiable if the difference quotient has a limit approaching the diagonal

$$isDifferentiable(f) := hasLimitApproachingDiagonal\left(\frac{f(x) - f(y)}{x - y}\right)$$

Let us define the type of differentiable functions $D^1(I, F)$ on $I$ as the type of pointwise continuous functions that are differentiable.

$$D^1(I, F) := \{f \in C^0(I, F) \vert isDifferentiable(f)\}$$

The Newton–Leibniz operator or D operator $\tilde{D}: D^1(I, F) \to (I \to F)$ is a function from the type of differentiable functions $D^1(I, F)$ to the type of functions $I \to F$, and is pointwise defined as

$$\tilde{D}(f) \coloneqq \lim_{(x, y) \to (x, x)} \frac{f(x) - f(y)}{x - y}$$

for a differentiable function $f:D^1(I, F)$. For any differentiable function $f:D^1(I, F)$, the function $\tilde{D}(f)$ is called the derivative of $f$.

Properties

These all follow from the algebraic limit theorems of the limit of a binary function approaching a diagonal.

Constant function rule

Let $c:I \to F$ be a constant function. With constant functions the function $c:I \to F$ and the value of the function $c:F$ are usually written with the same symbol. For all constant functions $c:I \to F$,

$$\tilde{D}(c) = \lim_{(x, y) \to (x, x)} \frac{c - c}{x - y}$$

$$\tilde{D}(c) = \lim_{(x, y) \to (x, x)} \frac{0}{x - y}$$

$$\tilde{D}(c) = \lim_{(x, y) \to (x, x)} 0$$

$$\tilde{D}(c) = 0$$

Identity function rule

Where $\frac{x}{x}$ is another notation for $x \cdot x^{-1}$

$$\tilde{D}(id_F) = \lim_{(x, y) \to (x, x)} \frac{x - y}{x - y}$$

$$\tilde{D}(id_F) = 1$$

Linearity

For all elements $a \in F$ and $b \in F$ and functions $f:I \to F$ and $g:I \to F$,

$$\tilde{D}(a \cdot f + b \cdot g) = \lim_{(x, y) \to (x, x)} \frac{(a \cdot f(x) + b \cdot g(x)) - (a \cdot f(y) + b \cdot g(y)}{x - y}$$

$$\tilde{D}(a \cdot f + b \cdot g) = \lim_{(x, y) \to (x, x)} \frac{(a \cdot f(x) - a \cdot f(y)) + (b \cdot g(x) - b \cdot g(y)}{x - y}$$

$$\tilde{D}(a \cdot f + b \cdot g) = \lim_{(x, y) \to (x, x)} \frac{(a \cdot (f(x) - f(y)) + (b \cdot (g(x) - g(y))}{x - y}$$

$$\tilde{D}(a \cdot f + b \cdot g) = \lim_{(x, y) \to (x, x)} \frac{a \cdot (f(x) - f(y)}{x - y} + \frac{b \cdot (g(x) - g(y)}{x - y}$$

$$\tilde{D}(a \cdot f + b \cdot g) = a \cdot \lim_{(x, y) \to (x, x)} \frac{(f(x) - f(y)}{x - y} + b \cdot \lim_{(x, y) \to (x, x)} \frac{(g(x) - g(y)}{x - y}$$

$$\tilde{D}(a \cdot f + b \cdot g) = a \cdot \tilde{D}(f) + b \cdot \tilde{D}(g)$$

Leibniz rule

For all functions $f:I \to F$ and $g:I \to F$,

$$\tilde{D}(f \cdot g) = \lim_{(x, y) \to (x, x)} \frac{(f(x) \cdot g(x)) - (f(y) \cdot g(y)}{x - y}$$

$$\tilde{D}(f \cdot g) = \lim_{(x, y) \to (x, x)} \frac{(f(x) \cdot g(x)) - (f(x) \cdot g(y)) + (f(x) \cdot g(y)) - (f(y) \cdot g(y)}{x - y}$$

$$\tilde{D}(f \cdot g) = \lim_{(x, y) \to (x, x)} \frac{(f(x) \cdot (g(x) - g(y)) + ((f(x) - f(y)) \cdot g(y))}{x - y}$$

$$\tilde{D}(f \cdot g) = \lim_{(x, y) \to (x, x)} \frac{f(x) \cdot (g(x) - g(y)}{x - y} + \frac{(f(x) - f(y)) \cdot g(y)}{x - y}$$

$$\tilde{D}(f \cdot g) = f \cdot \left(\lim_{(x, y) \to (x, x)} \frac{g(x) - g(y)}{x - y}\right) + \left(\lim_{(x, y) \to (x, x)} \frac{f(x) - f(y)}{x - y}\right) \cdot g$$

$$\tilde{D}(f \cdot g) = f \cdot \tilde{D}(g) + \tilde{D}(f) \cdot g$$

Power rule

For all functions $f:I \to F$ and natural numbers $n \in \mathbf{N}$,

$$\tilde{D}(f^n) = \lim_{(x, y) \to (x, x)} \frac{f(x)^n - f(y)^n}{x - y}$$

We proceed by induction on the natural numbers. The base case $n = 0$: For all functions $f:I \to F$,

$$\tilde{D}(f^0) = \lim_{(x, y) \to (x, x)} \frac{f(x)^0 - f(y)^0}{x - y}$$

$$\tilde{D}(f^0) = \lim_{(x, y) \to (x, x)} \frac{1 - 1}{x - y}$$

$$\tilde{D}(f^0) = \tilde{D}(1)$$

Then the inductive case: For all functions $f:I \to F$ and natural numbers $n \in \mathbf{N}$,

$$\tilde{D}(f^{n+1}) = \lim_{(x, y) \to (x, x)} \frac{f(x)^{n+1} - f(y)^{n+1}}{x - y}$$

$$\tilde{D}(f^{n+1}) = \lim_{(x, y) \to (x, x)} \frac{f(x) \cdot f(x)^{n} - f(y) \cdot f(y)^{n}}{x - y}$$

$$\tilde{D}(f^{n+1}) = \tilde{D}(f) \cdot f^n + \tilde{D}(f^n) \cdot f$$

$$\tilde{D}(f^{n+1}) = \sum_{n=0}^{n} \tilde{D}(f) \cdot f^n$$

Reciprocal rule

See also

• differentiable function

• iterated differentiable function

• smooth function

• antiderivative