Sample the Slopes and These Four Results Draw Themselves

You'll learnto read the derivatives of sine, cosine, e to the x and the natural log off their own graphs — including which member of the trigonometric pair carries the minus sign, and why only base e is its own slope.

Four functions no algebraic rule reaches, and four results to know cold. Two of them differ by one minus sign; two of them differ by a factor of a logarithm. Nearly every error in this topic is one of those two small differences going missing — which is exactly why nothing here is quoted at you. The slope curves in this file are measured off the plotted functions themselves, point by point, so you can watch sine's slopes land on the cosine and watch the minus sign appear at the place where the curve turns downward. Then the same treatment for the exponentials: draw the tangent to eˣ anywhere at all and its run back to the axis is exactly 1, which is another way of saying that its slope equals its height. Do it to 2ˣ and the run is 1.44 instead. That 1.44 is 1 / ln 2, and its refusal to be 1 is the entire reason base e is worth defining.

8 STEPS · 6 QUICK CHECKS · THE MINUS SIGN LIVES ON THE COSINE'S DERIVATIVE · RADIANS ONLY · v1

sin x ⇄ cos x · eˣ ⇄ eˣ · ln x ⇄ 1 / x

Before you start

What you're looking at

One period of sin x, with a tangent that will ride along it. Underneath is an empty panel waiting for the slope of that tangent, measured off the curve and dropped in as a dot at every x.

The question

When those measured slopes are joined up, which function do they draw — and where exactly does the minus sign that separates sine from cosine come from?

Watch for

The dots landing on a curve you already know, the slope trace going negative at the moment the curve above starts falling, and the tangent to eˣ meeting the axis exactly one unit back.

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