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VECTOR Calc 1–4
4d Streak
120 XP
Mode A: Guided Applet
x₀ = 1.00| f(x₀) = 0.750| f'(x₀) = 0.250
Dynamic Parameters
Target Point x₀ (x0x_0)1.0
Epsilon Corridor ε (ε\varepsilon)0.50
Delta Window δ (δ\delta)0.35
Visual Overlays & Layers
Chapter 3 Theory

Product, Quotient & Chain Rules

Decomposing composite functions and rates of change

Examine composite functions f(g(x)) and watch how inner derivatives gear-mesh with outer slopes to amplify the total rate of change.

1. The Chain Rule Multiplication of Rates

When one variable depends on another through an intermediate variable, their sensitivities multiply: dy/dx = (dy/du) * (du/dx). Geometrically, this acts like nested gear teeth transferring rotational velocity.

ddx[sin⁡(u(x))]=cos⁡(u(x))⋅u′(x)\frac{d}{dx}[\sin(u(x))] = \cos(u(x)) \cdot u'(x)

Quick Reference & Cheat Sheet

Product & Quotient Rules
(uv)′=u′v+uv′,(uv)′=u′v−uv′v2(uv)' = u'v + uv', \qquad \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}
Area expansion of a rectangle u×v and fractional rate of change.