Four Fractions, One Rule, and Every Co-Function Keeps the Minus

You'll learnto build all four remaining trigonometric derivatives out of the quotient rule, and to know which three of them carry a minus sign without memorising a table.

tan x is sin x over cos x, so it is a fraction, and you already have a rule for fractions. Run it and something small happens that shrinks the whole answer: the rule subtracts, the derivative of cos x is itself negative, and the two minus signs make a plus. The numerator becomes cos²x + sin²x, which is 1, and four terms of work collapse to sec²x. Do the same to cos x over sin x, to 1 over cos x and to 1 over sin x, and the other three fall out with one pattern holding every sign in place. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · FOUR QUOTIENT-RULE EXERCISES, NOT FOUR NEW FACTS · THE CO- NAMES CARRY THE MINUS · RADIANS ONLY · v2

tan x = sin x / cos x, so 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.

The question

Four derivatives are coming. Must each be memorised on its own, or does one rule you already have produce all four, signs included?

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