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Mathematics · 术语表

什么是 Derivative at a point？

又称： derivative

Definition 12.3 High School Mathematics · 第 12 章 — Differentiation: A First Course

Let $f$ be defined on an interval containing $a$ . For $h \neq 0$ , form the average rate of change between $a$ and $a + h$ :

r(h) = \frac{f(a+h) - f(a)}{h}.

If $r(h)$ approaches a single fixed number as $h$ gets arbitrarily close to $0$ , we say that $f$ is differentiable at $a$ ; this number is the derivative of $f$ at $a$ , written $f'(a)$ :

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

示例

Example 12.4

Let $f(x) = x^2$ and $a = 1$ . For $h \neq 0$ :

\frac{f(1+h) - f(1)}{h} = \frac{(1+h)^2 - 1}{h} = \frac{1 + 2h + h^2 - 1}{h} = \frac{h(2 + h)}{h} = 2 + h .

As $h$ approaches $0$ , this approaches $2$ : the function $x^2$ is differentiable at $1$ and $f'(1) = 2$ . Note the strategy: simplify the quotient until $h$ no longer appears in a denominator, then let $h \to 0$ .

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Definition 22.1 High School Mathematics · 第 22 章 — Differentiation and Convexity

Let $f$ be defined on an interval $I$ and $a \in I$ . The function $f$ is differentiable at $a$ if the difference quotient

\frac{f(a+h) - f(a)}{h}

has a finite limit as $h \to 0$ . This limit is called the derivative of $f$ at $a$ and written $f'(a)$ . If $f$ is differentiable at every point of $I$ , the function $f' \colon x \mapsto f'(x)$ is the derivative of $f$ .

示例

Example 22.15 (Convexity inequalities)

The exponential is convex on $\R$ (its second derivative is itself, positive). Its tangent at $0$ is $y = 1 + x$ , so

\eu^x \geq 1 + x \quad \text{for all } x \in \R .

Such tangent line inequalities are a standard product of convexity.

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