Differentiation

The gradient of a curve at a point: what a derivative means, the rules for finding one, and the three questions — tangents, stationary points and rates of change — that every calculus question on this paper reduces to.

หลักสูตรแกนกลาง / สสวท.: แคลคูลัส (Calculus)ตรงกับ ม.6 อนุพันธ์ของฟังก์ชัน (เพิ่มเติม) — 0606 เพิ่มอนุพันธ์ของฟังก์ชันตรีโกณมิติ เอกซ์โพเนนเชียล และลอการิทึม
Common Core (US): —
Cambridge: IGCSE 0606 — Topic 14: Differentiation and integration

11.1 What a derivative is

The gradient of a straight line is one number. A curve does not have one gradient — it has a different one at every point — so the derivative is not a number, it is a FUNCTION that hands you the gradient once you say where.

นิยาม

The derivative

$\frac{d y}{d x}$ is the gradient of the curve at the point $x$ . It is found as the limit of the gradient of a chord as the two ends slide together — which is why it is exactly the gradient of the tangent.

The rules this syllabus needs

\frac{d}{d x} (x^{n}) = n x^{n - 1} \frac{d}{d x} (sin x) = cos x \frac{d}{d x} (e^{x}) = e^{x} \frac{d}{d x} (ln x) = \frac{1}{x}

The trigonometric ones hold in RADIANS only — this is the promise chapter 7 made and the place it is cashed in.

Chain, product and quotient

\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}}

The quotient rule subtracts, so its order matters: the derivative of the TOP comes first. The product rule adds, so its order does not.

ตัวอย่าง 1

Differentiate $y = (3 x^{2} - 1)^{5}$ .

วิธีทำ

1.Let $u = 3 x^{2} - 1$ , so $y = u^{5}$
2. $\frac{d y}{d u} = 5 u^{4}$ and $\frac{d u}{d x} = 6 x$
3.Multiply them

ตอบ $\frac{d y}{d x} = 30 x (3 x^{2} - 1)^{4}$

ตัวอย่าง 2

Differentiate $y = x^{2} ln x$ .

วิธีทำ

1.Product rule with $u = x^{2}$ and $v = ln x$
2. $u^{'} = 2 x$ and $v^{'} = \frac{1}{x}$
3. $2 x ln x + x^{2} \cdot \frac{1}{x}$

ตอบ $\frac{d y}{d x} = 2 x ln x + x$

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