Introduction to Differential Calculus

What Is Differential Calculus?

Differential Calculus (single variable)

Differential calculus studies how a function changes at and near a point. It focuses on derivatives $f'(x)$, linear approximations $df = f'(x)\,dx$, and how these ideas explain tangents, extrema, and local behavior.

A brief origin story

The subject was shaped independently by Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century. Their work on motion, geometry, and algebra of infinitesimals gave us the derivative as a unifying tool for change.

Core questions we answer

1. Rate of change: How fast does $f(x)$ move when $x$ nudges by $\Delta x$?
2. Tangent slope: What is the slope of the curve at a single point?
3. Extrema: Where are the peaks and valleys, and how do we classify them?
4. Linear approximation: How do we replace a curved function with a straight line nearby?

How this course is organized

• Start with intuition (geometry and physics), then prove theorems (mean value, L’Hôpital).
• Practice the core derivative rules until they feel automatic.
• Apply derivatives to monotonicity, concavity, optimization, and error estimation.

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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
$f'(x)$	数学符号	“f prime of x”	函数在点 $x$ 的瞬时变化率
$dx$	数学符号	“dee x”	自变量的微小增量，用于线性近似
$\Delta x$	数学符号	Delta x（德尔塔）	自变量的有限增量，用于平均变化率

中英对照

中文术语	英文术语	音标	说明
一元函数	single-variable function	/ˈsɪŋɡl̩ ˈvɛərɪəbl ˈfʌŋkʃən/	只有一个自变量的函数
微分学	differential calculus	/ˌdɪfəˈrɛnʃəl ˈkælkjələs/	研究导数与微分的数学分支
导数	derivative	/dɪˈrɪvətɪv/	变化率或切线斜率
线性近似	linear approximation	/ˈlɪniə əˌprɒksɪˈmeɪʃən/	用切线逼近原函数
极值	extremum	/ɪkˈstriːməm/	函数的极大值或极小值

NextBasic Concepts of the Derivative