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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 2 Theory

The Derivative & Instantaneous Rate of Change

Secant lines collapsing into tangent lines as Δx → 0

Drag Δx towards 0 to see the secant chord rotate and snap directly onto the tangent line, revealing the derivative f'(x₀).

1. The Tangent as a Limit of Secants

The average rate of change over an interval [x₀, x₀ + Δx] is the slope of the secant line. Taking the limit as Δx → 0 yields the instantaneous rate of change—the slope of the tangent line.

mtangent=lim⁡h→0f(x0+h)−f(x0)hm_{\text{tangent}} = \lim_{h \to 0} \frac{f(x_0+h) - f(x_0)}{h}

Quick Reference & Cheat Sheet

Power Rule
ddx[xn]=nxn−1\frac{d}{dx}[x^n] = n x^{n-1}
Dilation of power curves under infinitesimal increments.