CAL-01

Power-rule derivative

d/dx x^n = n x^{n−1}. Instantaneous slope of a monomial.

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DerivativesPower rule

Governing equation

\dfrac{d}{dx}x^n=n x^{n-1}

$x \in \left[0.1, 10\right]$

where

n: Exponent (—)
x: x (—)
y': 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

\dfrac{d}{dx}x^n=n x^{n-1}

$x \in \left[0.1, 10\right]$

$x = 0.1$ $y' = 0.03$

Along the curve

②

Domain of the variable — blue

x_{\min} = 0.1,\quad x_{\max} = 10

$x \in \left[0.1, 10\right]$

$x = 0.1$ $x = 0.1$

Along the curve

③

Response curve — green

\dfrac{d}{dx}x^n=n x^{n-1}

$x \in \left[0.1, 10\right]$

$x = 0.1$ $y' = 0.03$

Along the curve

④

Rate of change — amber

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

$x \in \left[0.1, 10\right]$

$x = 0.162$ $y'^{\prime} = 0.974$

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-01 — Power-rule derivative) is the form associated with Power rule. Working symbols: $n$ , $x$ $\rightarrow$ $y'$ . From the binomial expansion of (x+h)^n, or from ln differentiation for real n.

Purpose

Purpose: compute

y'

from

n

x

in Calculus via

\dfrac{d}{dx}x^n=n x^{n-1}

d/dx x^n = n x^{n−1}. Instantaneous slope of a monomial. 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

\dfrac{d}{dx}x^n=n x^{n-1}

yields

y' = 12.0000\,\mathrm{—}

. Evaluate the derivative at a point x. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Exponent n3 —x x2 —

Outputs

Derivative y'12.0000 —

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Narration of this film

Evaluate the derivative at a point x.

From the binomial expansion of (x+h)^n, or from ln differentiation for real n.

Reading speed

Watch on YouTube