CAL-10

Implicit derivative snapshot

F(x,y) = c ⇒ dy/dx = −Fx / Fy. Slope of a level curve.

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DerivativesImplicit differentiation

Governing equation

\dfrac{dy}{dx}=-\dfrac{F_x}{F_y}

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

where

F_x: ∂F/∂x (—)
F_y: ∂F/∂y (—)
y': Implicit slope (—)

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{dy}{dx}=-\dfrac{F_x}{F_y}

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

$F_x = -20$ $y' = 5$

Along the curve

②

Domain of the variable — blue

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

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

$F_x = -20$ $F_x = -20$

Along the curve

③

Response curve — green

\dfrac{dy}{dx}=-\dfrac{F_x}{F_y}

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

$F_x = -20$ $y' = 5$

Along the curve

④

Rate of change — amber

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

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

$F_x = -19.75$ $y'^{\prime} = -0.25$

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-10 — Implicit derivative snapshot) is the form associated with Implicit differentiation. Working symbols: $F_x$ , $F_y$ $\rightarrow$ $y'$ . Chain rule on F(x, y(x)) = c. The gradient is orthogonal to the level set.

Purpose

Purpose: compute

y'

from

F_x

F_y

in Calculus via

\dfrac{dy}{dx}=-\dfrac{F_x}{F_y}

F(x,y) = c ⇒ dy/dx = −Fx / Fy. Slope of a level curve. 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_x = 2.000\,\mathrm{—}

F_y = 4.000\,\mathrm{—}

, the governing relation

\dfrac{dy}{dx}=-\dfrac{F_x}{F_y}

yields

y' = -0.5000\,\mathrm{—}

. Enter partials Fx, Fy at a point on the curve. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

∂F/∂x F_x2 —∂F/∂y F_y4 —

Outputs

Implicit slope y'-0.5000 —

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Narration of this film

Enter partials Fx, Fy at a point on the curve.

Chain rule on F(x, y(x)) = c. The gradient is orthogonal to the level set.

Reading speed

Watch on YouTube