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MathMagic

Derivatives as Slope

Start from the slope of a line, squeeze two points together, and watch the tangent and the derivative appear.

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video16:50 · 8 chapters

The paradox of the derivative | Essence of calculus, chapter 2

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chapters

step 01 / 04 · Slope between two points

plotting…

The curve y equals x squared with a line through the points at x 0.5 and 1.5.

0 of 4 understood

  1. Slope between two points

    On a straight line the slope is rise over run. A curve bends, so pick two points on it and draw the line through them, a secant.

    For f(x)=x2f(x)=x^2 at x=0.5x=0.5 and x=1.5x=1.5:

    f(1.5)−f(0.5)1.5−0.5=2.25−0.251=2\frac{f(1.5)-f(0.5)}{1.5-0.5} = \frac{2.25-0.25}{1} = 2

  2. Squeeze the second point closer

    Slide the second point toward the first. With a gap hh the secant slope is

    (x+h)2−x2h=2x+h\frac{(x+h)^2 - x^2}{h} = 2x + h

    At x=0.5x=0.5 and h=0.3h=0.3 that is 1.31.3, already close to 11.

  3. In the limit: the tangent

    Let hh shrink to zero. The secant settles into a single line touching the curve at one point, the tangent, and its slope is the limit

    f′(x)=lim⁡h→0f(x+h)−f(x)h=2xf'(x) = \lim_{h\to 0}\frac{f(x+h)-f(x)}{h} = 2x

    At x=0.5x = 0.5 the slope is exactly 11.

  4. The slope is itself a function

    Move the point to x=1.5x=1.5 and the slope becomes 33. Do this everywhere and the slopes trace out a new curve, f′(x)=2xf'(x)=2x, drawn dashed.

    That is the derivative: a function that reports the slope of ff at each point.