Basic Concepts of the Derivative

The derivative measures an instantaneous rate of change—how fast $f(x)$ moves when $x$ moves by a tiny amount.

Average vs. instantaneous rate

Average rate on $[x_0, x_0+\Delta x]$:

$$\frac{\Delta y}{\Delta x} = \frac{f(x_0+\Delta x) - f(x_0)}{\Delta x}.$$

Instantaneous rate (the derivative):

$$f'(x_0) = \lim_{\Delta x \to 0} \frac{f(x_0+\Delta x) - f(x_0)}{\Delta x}.$$

Geometric picture

Average rate = secant slope; instantaneous rate = tangent slope. Explore this with $f(x)=x^2$:

Formal definition

Why $\dfrac{dy}{dx}$?

• $\dfrac{\Delta y}{\Delta x}$: slope of a secant (finite increments).
• $\dfrac{dy}{dx}$: slope of the tangent (infinitesimal increments).

Equivalent forms

$$f'(x_0) = \lim_{h\to 0} \frac{f(x_0+h)-f(x_0)}{h} \quad\text{or}\quad f'(x_0) = \lim_{x\to x_0} \frac{f(x)-f(x_0)}{x-x_0}.$$

One-sided derivatives

• Left: $f'_-(x_0) = \lim_{h\to 0^-} \dfrac{f(x_0+h)-f(x_0)}{h}$
• Right: $f'_+(x_0) = \lim_{h\to 0^+} \dfrac{f(x_0+h)-f(x_0)}{h}$

Differentiable at $x_0$ ⇔ both exist and are equal.

Differentiability vs. continuity

Examples and practice

Example: $f(x)=x^2$

$$f'(x_0) = \lim_{\Delta x\to 0} \frac{(x_0+\Delta x)^2 - x_0^2}{\Delta x} = 2x_0.$$

练习 1

Differentiate $f(x) = \dfrac{1}{x}$ at $x_0$ using the definition.

练习 2

Is $f(x)=|x|$ differentiable at $0$?

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Summary

本文出现的符号

中英对照