M171 Calculus I · Unit 2: Computing derivatives · on Exam 2
§2.4 Derivatives of other trigonometric functions
Study notes for Derivatives of other trigonometric functions in M171 Calculus I at Montana State University: the big ideas, key formulas, a worked example, common mistakes and an exam tip. The same topic has endless practice questions and flashcards in Mathub.
Big ideas
- All four come from the quotient rule applied to $\tan x = \frac{\sin x}{\cos x}$, $\cot x = \frac{\cos x}{\sin x}$, $\sec x = \frac1{\cos x}$, $\csc x = \frac1{\sin x}$.
- Pattern: every "co" function has a negative derivative.
Key formulas
tan / cot
$$\frac{d}{dx}[\tan x] = \sec^2 x,\qquad \frac{d}{dx}[\cot x] = -\csc^2 x$$
sec / csc
$$\frac{d}{dx}[\sec x] = \sec x\tan x,\qquad \frac{d}{dx}[\csc x] = -\csc x\cot x$$
Worked example
Find $f'(x)$ for $f(x) = x\tan x + 5\sec x$.
Show the solution
$f'(x) = \tan x + x\sec^2 x + 5\sec x\tan x$.
Common mistakes
- Writing $\frac{d}{dx}[\tan x] = \sec x$ (missing the square).
Exam tipIf you blank on one of these, rederive it from the quotient rule in two lines; e.g. $\frac{d}{dx}\frac{\sin x}{\cos x} = \frac{\cos^2x + \sin^2x}{\cos^2x} = \sec^2x$.
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