Differentiation (Pure 1)

The derivative as a limit, the rules that follow from it, and the three things Pure 1 asks you to do with one: tangents and normals, stationary points, and connected rates of change.

หลักสูตรแกนกลาง / สสวท.: แคลคูลัส (Calculus)บทนี้เปิดด้วยนิยามลิมิตของอนุพันธ์ ซึ่งตรงกับ ม.6 ลิมิตและความต่อเนื่อง (เพิ่มเติม) แล้วต่อไปยังกฎการหาอนุพันธ์
Common Core (US): —
Cambridge: 9709 Pure Mathematics 1 — Differentiation

5.1 The derivative as a limit

A chord joining two points on a curve has a gradient you can calculate. Slide the two points together and that gradient settles on a value — the gradient of the tangent. The derivative is the name for that limit, and every rule in this chapter is a consequence of it.

First principles

\frac{d y}{d x} = h \to 0 lim \frac{f ( x + h ) - f ( x )}{h}

The $h$ in the denominator is why this needs a limit rather than a substitution: at $h = 0$ the fraction is $\frac{0}{0}$ , which says nothing.

ตัวอย่าง

Differentiate $f (x) = x^{2}$ from first principles.

วิธีทำ

1. $\frac{( x + h ) ^{2} - x ^{2}}{h} = \frac{2 x h + h ^{2}}{h}$
2.For $h \neq = 0$ this simplifies to $2 x + h$
3.As $h \to 0$ it approaches $2 x$

ตอบ $f^{'} (x) = 2 x$

The rules

\frac{d}{d x} (x^{n}) = n x^{n - 1} \frac{d y}{d x} = \frac{d y}{d u} \frac{d u}{d x} (uv)^{'} = u^{'} v + u v^{'} (\frac{u}{v})^{'} = \frac{u ^{'} v - u v ^{'}}{v ^{2}}

Before differentiating, rewrite roots and fractions as powers: $x$ is $x^{1/2}$ and $\frac{1}{x ^{3}}$ is $x^{- 3}$ . The power rule then covers all of them.

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