CAL-02

Product rule

(uv)' = u' v + u v'. Snapshot at a point.

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DerivativesProduct rule

Governing equation

(uv)'=u'v+uv'

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

where

u: u (—)
u': u' (—)
v: v (—)
v': v' (—)
(uv)': Product 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

(uv)'=u'v+uv'

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

$u = -20$ $(uv)' = 21.5$

Along the curve

②

Domain of the variable — blue

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

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

$u = -20$ $u = -20$

Along the curve

③

Response curve — green

(uv)'=u'v+uv'

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

$u = -20$ $(uv)' = 21.5$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}(uv)'}{\mathrm{d}u}

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

$u = -19.75$ $(uv)'^{\prime} = -1$

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-02 — Product rule) is the form associated with Product rule. Working symbols: $u$ , $u'$ , $v$ , $v'$ $\rightarrow$ $(uv)'$ . The extra area of a growing rectangle is the two strips u'v Δx and uv' Δx.

Purpose

Purpose: compute

(uv)'

from

u

u'

v

v'

in Calculus via

(uv)'=u'v+uv'

(uv)' = u' v + u v'. Snapshot at a point. 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 = 2.000\,\mathrm{—}

u' = 0.500\,\mathrm{—}

v = 3.000\,\mathrm{—}

v' = -1.000\,\mathrm{—}

, the governing relation

(uv)'=u'v+uv'

yields

(uv)' = -0.5000\,\mathrm{—}

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

Calculator

Inputs

u u2 —u' u'0.5 —v v3 —v' v'-1 —

Outputs

Product derivative (uv)'-0.5000 —

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Narration of this film

Enter u, v and their derivatives at the same x.

The extra area of a growing rectangle is the two strips u'v Δx and uv' Δx.

Reading speed

Watch on YouTube