Every Co-Function Carries the Minus, and the Quotient Rule Says Why

You'll learnto produce all four remaining trigonometric derivatives from the quotient rule, and to keep each one in its own family with the sign it is owed.

tan x is sin x over cos x. That is not a new kind of object — it is a quotient, and Topic 2.9 already said what to do with one. Run the quotient rule on it and something small happens that changes the whole answer: the rule's subtraction meets the minus sign that the derivative of cos x carries, the two combine into a plus, and the numerator becomes cos²x + sin²x — which the Pythagorean identity flattens to 1. Four terms of work collapse into 1/cos²x, that is, sec²x. Do the same to cos x / sin x, to 1/cos x and to 1/sin x, and the other three fall out, with one pattern holding all four signs in place: the three functions whose names begin with co — cosine, cotangent, cosecant — are exactly the three whose derivatives are negative. Not a table to memorise. A symmetry, and a picture that checks it.

8 STEPS · 6 QUICK CHECKS · FOUR QUOTIENT-RULE EXERCISES, NOT FOUR NEW FACTS · EVERY CO-FUNCTION CARRIES THE MINUS · RADIANS ONLY · v1

tan x = sin x / cos x ⇄ d/dx[tan x] = sec²x

Before you start

What you're looking at

The graph of tan x, in radians, drawn one branch at a time. The dashed verticals mark the inputs where cos x = 0 and tan x has no value at all.

The question

Four derivatives are coming. Do they have to be memorised separately — or does one rule you already own produce all four, signs included?

Watch for

Two waves whose heights always add to 1, tan and cot drawn as mirror images, and the moment a dropped minus sign puts a rising tangent line on a falling curve.

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