Secant Becomes Tangent

Also called

secant to tangent · tangent line · slope of a curve · derivative · first principles · derivative from first principles · limit of a difference quotient · rise over run · instantaneous rate of change · derivative of x squared · left and right limits · modulus function · not differentiable at a corner · स्पर्श रेखा · अवकलज · सीमा

Topics

• The secant through P(x, f(x)) and Q(x + h, f(x + h)) has slope (f(x + h) − f(x)) / h: rise over run.
• As Q slides towards P (h → 0) the secant turns and settles on one line: the tangent at P.
• f′(x) = lim (h → 0) of (f(x + h) − f(x)) / h, the slope of the tangent: the derivative from first principles.
• For f(x) = x² the secant slope is 2x + h, which settles on 2x: the derivative of x² is 2x.
• At x = ½ the slope settles on 1 whether Q comes in from the right or from the left: both one-sided limits agree.
• Plotting the tangent’s slope at every x draws the derivative: for x² it is the straight line y = 2x.
• At the bottom of the parabola the tangent is level, so the derivative is zero there.
• y = |x| has a corner at 0: the slopes from the right (+1) and left (−1) differ, so it has no derivative there.
• A function differentiable at a point is continuous there, but |x| shows that being continuous is not enough.

About this model

A GLB file of about 290 KB, shown in Lably's 3D viewer and, on a phone, placed in your room with AR. Credit: Model, science and animation: Lably (built procedurally with three.js; no downloaded meshes). | CC BY 4.0. Original geometry and animation made for Lably.

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