Leibniz and the Rise of Differentials

A geometric puzzle

Leibniz, 1675: How do we draw the tangent to any curve? He experiments with the parabola $y=x^2$, comparing secant slopes as two points approach each other.

For points $(x,x^2)$ and $(x+\Delta x,(x+\Delta x)^2)$, the secant slope is

$$\frac{(x+\Delta x)^2 - x^2}{\Delta x} = 2x + \Delta x \to 2x \quad (\Delta x \to 0).$$

The limit is the tangent slope.

Differentials

Leibniz introduces differentials to formalize this limit:

$$\frac{dy}{dx} = \lim_{\Delta x \to 0} \frac{f(x+\Delta x) - f(x)}{\Delta x}.$$

• $dx$: an infinitesimal change of $x$
• $dy$: the corresponding infinitesimal change of $y$
• $\dfrac{dy}{dx}$: their ratio, the derivative

Rules that survived

• Power rule: $(x^n)' = nx^{n-1}$
• Exponential: $(e^x)' = e^x$
• Logarithm: $(\ln x)' = \dfrac{1}{x}$
• Product: $d(uv) = u\,dv + v\,du$
• Quotient: $d\!\left(\dfrac{u}{v}\right) = \dfrac{v\,du - u\,dv}{v^2}$
• Chain rule: $dy = \dfrac{dy}{du}\cdot \dfrac{du}{dx}\;dx$

These notations and rules, polished by later mathematicians, became the standard calculus language.

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总结

本文出现的符号

符号	类型	读音/说明	在本文中的含义
$dx, dy$	符号	d x, d y	自变量与函数的无穷小变化量
$\frac{dy}{dx}$	符号	d y over d x	切线斜率（导数）

中英对照

中文术语	英文术语	音标	说明
微分	differential	/ˌdɪfəˈrɛnʃəl/	用无穷小量刻画变化
导数	derivative	/dɪˈrɪvətɪv/	变化率或切线斜率
割线	secant line	/ˈsiːkənt laɪn/	连接曲线上两点的直线
切线	tangent line	/ˈtændʒənt laɪn/	与曲线在一点相切的直线
链式法则	chain rule	/tʃeɪn ruːl/	复合函数的求导规则

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