Derivatives Field Guide

Trig derivatives: learn 3, get 6

Don't memorize six separate facts. Learn the three “plain” ones, then use one rule to get the other three for free.

Step 1. Learn these three:

Function	Derivative
$\sin x$	$\cos x$
$\tan x$	$\sec^2 x$
$\sec x$	$\sec x\tan x$

Step 2. The “co-” rule: put co in front of every function, then add a minus sign.

Plain	Add “co” + minus
$(\sin x)'=\cos x$	$(\cos x)'=-\sin x$
$(\tan x)'=\sec^2x$	$(\cot x)'=-\csc^2x$
$(\sec x)'=\sec x\tan x$	$(\csc x)'=-\csc x\cot x$

Adding “co” to cos gives “co-cos”, which is just sin. So in the first row, “cos x” becomes “sin x” and picks up the minus: $(\cos x)'=-\sin x$ .

The one sentence

“Co” functions get a minus. cos, cot and csc have negative derivatives. sin, tan and sec don't.

Partners

tan goes with sec, cot goes with csc. $\tan\to\sec^2$ , $\sec\to\sec\tan$ . $\cot\to-\csc^2$ , $\csc\to-\csc\cot$ .

Keep yourself

sec and csc keep themselves: $\sec\to\boldsymbol{\sec}\cdot\tan$ , $\csc\to-\boldsymbol{\csc}\cdot\cot$ .

The circle

sin and cos cycle every 4 derivatives: $\sin\to\cos\to-\sin\to-\cos\to\sin$ . One quarter turn per derivative.

Blanked on the exam? Rebuild tan in 20 seconds

You only need sin, cos and the quotient rule. Write $\tan x=\dfrac{\sin x}{\cos x}$ :

Low d-high minus high d-low: $\dfrac{\cos x\cdot\cos x-\sin x\cdot(-\sin x)}{\cos^2x}$
Top: $\cos^2x+\sin^2x=1$ .
So $(\tan x)'=\dfrac{1}{\cos^2x}=\sec^2x$ .
Same idea for sec: $\sec x=\dfrac1{\cos x}$ gives $\dfrac{\sin x}{\cos^2x}=\sec x\tan x$ .

Every rule as a phrase you can say

Rule	Formula	Say it as
Power	$(x^n)'=nx^{n-1}$	“Bring the power down, subtract one.”
Constant	$(c)'=0$	“Flat line, zero slope.”
$e^x$	$(e^x)'=e^x$	“e to the x is lazy: it stays the same.”
$a^x$	$(a^x)'=a^x\ln a$	“Same as e to the x, times ln a.”
Product	$(fg)'=fg'+gf'$	“First d-second, plus second d-first.”
Quotient	$\left(\frac fg\right)'=\frac{gf'-fg'}{g^2}$	“Low d-high minus high d-low, over the square of what's below.”
Chain	$f'(g(x))\,g'(x)$	“Outside derivative, inside left alone, times inside derivative.” Peel the onion.
Implicit	$(y^n)'=ny^{n-1}y'$	“Every y gets a y′.”

Why “low” goes first in the quotient rule: the minus sign makes order matter. Start with the bottom and you'll get the sign right.

Flashcard drill

Tap a card to flip it. Mark Got it or Again. Cards you miss come back at the end of the round. Switch the direction to practice going from the answer back to the question.

Answer first (reverse)

0 / 21 learned

Flip the card first, then mark it. On a keyboard: Space flips, → is Got it, ← is Again.

How to make it stick

Blank-page recall. Once a day, write every formula from memory on an empty sheet. Then check it against the cheat sheet. Recalling works much better than rereading.
Say the phrases out loud. They're written to be spoken. Rhythm helps memory.
Drill both directions. Use the flashcards above both ways: “ $\tan x\to$ ?” and “ $\sec^2x$ is the derivative of ?”.
Short and spread out. Ten minutes a day for several days beats one long night. Mix in a few problems from §3.3 and the Quick quiz.
When stuck, derive it. Rebuilding tan from sin/cos also practices the quotient rule.

Mixing up sec² and sec tan

tan's derivative is $\sec^2x$ . sec's derivative is $\sec x\tan x$ . Remember: sec keeps itself.

Dropping the minus on cos

$(\cos x)'=-\sin x$ . “Co” gets a minus, every time.

Power rule on aˣ or eˣ

$(2^x)'\ne x\,2^{x-1}$ . The power rule needs the variable in the base, not the exponent.