CAL-03

Chain rule

(f∘g)' = f'(g) g'. Composition of rates.

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DerivativesChain rule

Governing equation

(f\circ g)'=f'(g)\,g'

$f'(g) \in \left[-30, 30\right]$

where

f'(g): Outer derivative (—)
g': Inner derivative (—)
(f\circ g)': Chain 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

(f\circ g)'=f'(g)\,g'

$f'(g) \in \left[-30, 30\right]$

$f'(g) = -30$ $(f\circ g)' = -15$

Along the curve

②

Domain of the variable — blue

f'(g)_{\min} = -30,\quad f'(g)_{\max} = 30

$f'(g) \in \left[-30, 30\right]$

$f'(g) = -30$ $f'(g) = -30$

Along the curve

③

Response curve — green

(f\circ g)'=f'(g)\,g'

$f'(g) \in \left[-30, 30\right]$

$f'(g) = -30$ $(f\circ g)' = -15$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}(f\circ g)'}{\mathrm{d}f'(g)}

$f'(g) \in \left[-30, 30\right]$

$f'(g) = -29.62$ $(f\circ g)'^{\prime} = 0.5$

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-03 — Chain rule) is the form associated with Chain rule. Working symbols: $f'(g)$ , $g'$ $\rightarrow$ $(f\circ g)'$ . Leibniz: dy/dx = (dy/du)(du/dx). The Jacobian of a composition is a product of Jacobians.

Purpose

Purpose: compute

(f\circ g)'

from

f'(g)

g'

in Calculus via

(f\circ g)'=f'(g)\,g'

(f∘g)' = f'(g) g'. Composition of rates. Use it when a real calculus question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

f'(g) = 4.000\,\mathrm{—}

g' = 0.500\,\mathrm{—}

, the governing relation

(f\circ g)'=f'(g)\,g'

yields

(f\circ g)' = 2.0000\,\mathrm{—}

. Enter f'(g) and g' at the evaluation point. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Outer derivative f'(g)4 —Inner derivative g'0.5 —

Outputs

Chain derivative (f\circ g)'2.0000 —

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Narration of this film

Enter f'(g) and g' at the evaluation point.

Leibniz: dy/dx = (dy/du)(du/dx). The Jacobian of a composition is a product of Jacobians.

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