CAL-23

Power rule

Elementary derivative.

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Calculuspower

Governing equation

d/dx\\,x^n=n x^{n-1}

$x \in \left[-5, 5\right]$

where

n: n (—)
x: x (—)
f': f' (—)

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

d/dx\\,x^n=n x^{n-1}

$x \in \left[-5, 5\right]$

$x = -5$ $f' = -75$

Along the curve

②

Domain of the variable — blue

x_{\min} = -5,\quad x_{\max} = 5

$x \in \left[-5, 5\right]$

$x = -5$ $x = -5$

Along the curve

③

Response curve — green

d/dx\\,x^n=n x^{n-1}

$x \in \left[-5, 5\right]$

$x = -5$ $f' = -75$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}f'}{\mathrm{d}x}

$x \in \left[-5, 5\right]$

$x = -4.937$ $f'^{\prime} = 29.62$

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-23 — Power rule) is the form associated with power. Working symbols: $n$ , $x$ $\rightarrow$ $f'$ . Elementary derivative.

Purpose

Purpose: compute

f'

from

n

x

in Calculus via

d/dx\\,x^n=n x^{n-1}

Elementary derivative. Use it when a real calculus question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

n = 3.000\,\mathrm{—}

x = 2.000\,\mathrm{—}

, the governing relation

d/dx\\,x^n=n x^{n-1}

yields

f' = 12.0000\,\mathrm{—}

. Elementary derivative. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

n n3 —x x2 —

Outputs

f' f'12.0000 —

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Narration of this film

Elementary derivative.

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