Mathematics and Statistics · Ch 9 — Differentiation

Rules of Differentiation

Most functions are combinations of the standard ones, and four algebraic rules let us differentiate any such combination. Let $u=u(x)$ and $v=v(x)$ be differentiable, and $k$ a constant.

1. Constant-multiple rule:

\frac{d}{dx}\big[k\,u\big] = k\,\frac{du}{dx}.

2. Sum / difference rule:

\frac{d}{dx}\big[u \pm v\big] = \frac{du}{dx} \pm \frac{dv}{dx}.

Together these two rules let us differentiate any polynomial term by term.

3. Product rule — the derivative of a product is not the product of the derivatives:

\frac{d}{dx}\big[u\,v\big] = u\,\frac{dv}{dx} + v\,\frac{du}{dx} \qquad (\text{``first} \times \text{d(second)} + \text{second} \times \text{d(first)''}).

4. Quotient rule:

\frac{d}{dx}\!\left[\frac{u}{v}\right] = \frac{v\,\dfrac{du}{dx} - u\,\dfrac{dv}{dx}}{v^{2}}, \qquad v \neq 0 \qquad \left(\frac{\text{den}\times\text{d(num)} - \text{num}\times\text{d(den)}}{\text{den}^2}\right).

Illustration (product). For $y = x^2 \sin x$ : take $u=x^2$ ( $u'=2x$ ) and $v=\sin x$ ( $v'=\cos x$ ). Then $y' = x^2\cos x + \sin x\,(2x) = x^2\cos x + 2x\sin x$ .

Note

Order Matters in the Quotient Rule; Sign Does Not in the Product Rule …

Definition 10Sum, difference, constant-multiple rules

$\frac{d}{dx}(u\pm v)=u'\pm v'$ and $\frac{d}{dx}(ku)=ku'$ . Together they differentiate any polyno …

Definition 11Product rule

$\frac{d}{dx}(uv)=u\,v'+v\,u'$ . The derivative of a product is not the product of t …

Definition 12Quotient rule

$\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{v\,u'-u\,v'}{v^2}$ , $v\neq0$ . Denominator times derivative-of-numerator comes f …

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