CAL-21

Quotient rule snapshot

(u/v)' = (u' v − u v') / v².

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DerivativesQuotient rule

Governing equation

\left(\dfrac u v\right)'=\dfrac{u'v-uv'}{v^2}

$v \in \left[-20, 20\right]$

where

u: u (—)
u': u' (—)
v: v (—)
v': v' (—)
(u/v)': Quotient derivative (—)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

\left(\dfrac u v\right)'=\dfrac{u'v-uv'}{v^2}

$v \in \left[-20, 20\right]$

$v = -20$ $(u/v)' = -0.055$

Along the curve

②

Domain of the variable — blue

v_{\min} = -20,\quad v_{\max} = 20

$v \in \left[-20, 20\right]$

$v = -20$ $v = -20$

Along the curve

③

Response curve — green

\left(\dfrac u v\right)'=\dfrac{u'v-uv'}{v^2}

$v \in \left[-20, 20\right]$

$v = -20$ $(u/v)' = -0.055$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}(u/v)'}{\mathrm{d}v}

$v \in \left[-20, 20\right]$

$v = -19.75$ $(u/v)'^{\prime} = -0.003$

Along the curve

Lecture brief

Historical brief

Newton and Leibniz (1670s), Taylor, the fundamental theorem and the trapezoid rule are how change became a number. The lab differentiates, integrates and linearises in one variable. This sheet (CAL-21 — Quotient rule snapshot) is the form associated with Quotient rule. Working symbols: $u$ , $u'$ , $v$ , $v'$ $\rightarrow$ $(u/v)'$ . Product rule on u · v^{−1}, plus the chain rule on v^{−1}.

Purpose

Purpose: compute

(u/v)'

from

u

u'

v

v'

in Calculus via

\left(\dfrac u v\right)'=\dfrac{u'v-uv'}{v^2}

(u/v)' = (u' v − u v') / v². Use it when a real calculus question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

u = 4.000\,\mathrm{—}

u' = 1.000\,\mathrm{—}

v = 2.000\,\mathrm{—}

v' = 0.500\,\mathrm{—}

, the governing relation

\left(\dfrac u v\right)'=\dfrac{u'v-uv'}{v^2}

yields

(u/v)' = 0.0000\,\mathrm{—}

. Enter u, v and their derivatives at a point. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

u u4 —u' u'1 —v v2 —v' v'0.5 —

Outputs

Quotient derivative (u/v)'0.0000 —

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Narration of this film

Enter u, v and their derivatives at a point.

Product rule on u · v^{−1}, plus the chain rule on v^{−1}.

Reading speed

Watch on YouTube