Geometric Meaning of Derivatives

Derivatives connect algebra with geometry. Thinking of $f'(x)$ as a slope on the graph makes local behavior, monotonicity, and optimization much more intuitive.

Tangent slope

Basic idea

The derivative $f'(x_0)$ is the slope of the tangent line to $y = f(x)$ at the point $(x_0, f(x_0))$.

y = x² 及其切线

y = x²y = x³y = sin(x)y = e^x

图示展示了函数在特定点的切线。切线的斜率即为该点的导数值。

Tangent equation

Through $(x_0, f(x_0))$, the tangent line is

$$y - f(x_0) = f'(x_0)(x - x_0),$$

or equivalently $y = f(x_0) + f'(x_0)(x - x_0)$.

Normal equation

The normal line is perpendicular to the tangent, with slope $k_n = -\dfrac{1}{f'(x_0)}$:

$$y - f(x_0) = -\frac{1}{f'(x_0)}(x - x_0).$$

Note: if $f'(x_0) = 0$, the tangent is horizontal and the normal is vertical; if the derivative does not exist, the tangent is vertical.

Geometry vs. function behavior

Monotonicity and derivative sign

• $f'(x_0) > 0$: locally increasing.
• $f'(x_0) < 0$: locally decreasing.
• $f'(x_0) = 0$: candidate for extrema or plateau.

Extrema and slopes

At a local extremum, the tangent is typically horizontal:

1. Local maximum: $f'(x_0) = 0$ and the derivative changes from $+$ to $-$.
2. Local minimum: $f'(x_0) = 0$ and the derivative changes from $-$ to $+$.
3. Inflection candidate: $f'(x_0) = 0$ but no sign change.

Concavity

The second derivative $f''(x)$ reflects bending:

• $f''(x) > 0$: curve bends upward (convex/concave up).
• $f''(x) < 0$: curve bends downward (concave/convex down).

Uses of tangents

Linear approximation

$f(x) \approx f(x_0) + f'(x_0)(x - x_0)$

Good when $x$ is close to $x_0$.

Error estimation

$E(x) = f(x) - \big[f(x_0) + f'(x_0)(x - x_0)\big]$

As $x \to x_0$, the error $E(x)$ is higher order than $(x - x_0)$.

Optimization intuition

• Horizontal tangents mark potential extrema.
• The negative gradient points toward fastest descent.
• Newton’s method replaces the curve with its tangent to iterate quickly.

Special geometric cases

Non-differentiable points

1. Cusp: e.g., $f(x) = |x|$ at $x = 0$.
2. Vertical tangent: e.g., $f(x) = \sqrt[3]{x}$ at $x = 0$.
3. Oscillatory point: derivative fails due to rapid oscillation.

Horizontal tangents

If $f'(x_0) = 0$, possibilities include:

1. Extremum: local peak or valley.
2. Inflection: concavity changes while slope is zero.
3. Plateau: flat region with slow change.

练习题

练习 1

Find the tangent and normal lines of $f(x) = x^2$ at $(1, 1)$.

参考答案

1. $f'(x) = 2x$, so $f'(1) = 2$.
2. Tangent: $y - 1 = 2(x - 1)$ ⇒ $y = 2x - 1$.
3. Normal: $y - 1 = -\dfrac{1}{2}(x - 1)$ ⇒ $y = -\dfrac{1}{2}x + \dfrac{3}{2}$.

练习 2

Show that if $f'(x) > 0$ for all $x \in (a, b)$, then $f$ is strictly increasing on $(a, b)$.

参考答案

Let $x_1 < x_2$ in $(a, b)$. By the Mean Value Theorem, there exists $\xi \in (x_1, x_2)$ such that

$\xi$（Xi）：希腊字母，读作“克西”，常用来表示中值定理中的某一点。

$f(x_2) - f(x_1) = f'(\xi)(x_2 - x_1)$. Because $f'(\xi) > 0$ and $x_2 - x_1 > 0$, we get $f(x_2) > f(x_1)$, so $f$ is strictly increasing.

练习 3

Find all extrema of $f(x) = x^3 - 3x^2 + 2$ and classify them.

参考答案

1. $f'(x) = 3x^2 - 6x = 3x(x - 2)$ ⇒ critical points at $x = 0, 2$.
2. $f''(x) = 6x - 6$:
   • $f''(0) = -6 < 0$ ⇒ local maximum at $x = 0$, value $f(0) = 2$.
   • $f''(2) = 6 > 0$ ⇒ local minimum at $x = 2$, value $f(2) = -2$.

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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”	曲线在点 $x$ 处的切线斜率
$f''(x)$	数学符号	“f double prime of x”	二阶导数，刻画凹凸性
$\xi$	希腊字母	Xi（克西）	中值定理中的某一点

中英对照

中文术语	英文术语	音标	说明
切线	tangent line	/ˈtændʒənt laɪn/	与曲线在一点相切的直线
切线斜率	slope of tangent line	/sləʊp əv ˈtændʒənt laɪn/	切线在切点处的斜率，等于导数
法线	normal line	/ˈnɔːməl laɪn/	与切线垂直的直线
单调递增	monotonically increasing	/mɒnəˈtɒnɪkli ɪnˈkriːsɪŋ/	函数值随自变量增大而增大
单调递减	monotonically decreasing	/mɒnəˈtɒnɪkli dɪˈkriːsɪŋ/	函数值随自变量增大而减小
极值	extremum	/ɪkˈstriːməm/	函数在某点的极大值或极小值
拐点	inflection point	/ɪnˈflekʃən pɔɪnt/	函数改变凹凸性的点
凹凸性	concavity	/kənˈkævəti/	曲线的弯曲方向
线性近似	linear approximation	/ˈlɪniə əprɒksɪˈmeɪʃən/	用切线近似函数值
误差估计	error estimation	/ˈerə estɪˈmeɪʃən/	估计近似值的误差大小
牛顿法	Newton’s method	/ˈnjuːtənz ˈmeθəd/	利用切线进行迭代求解的方法
尖点	cusp	/kʌsp/	曲线上的尖锐转折点
不可导	non-differentiable	/nɒn dɪfəˈrenʃəbl/	函数在某点处导数不存在

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