Mathfolis

Trig Derivatives

Unit 2 · Differentiation — Rules

Derivatives of Trigonometric Functions

All six trig derivatives must be memorized. The 'co-' functions (cos, cot, csc) always carry a negative sign in their derivatives. These rules combine with the product, quotient, and chain rules for more complex expressions.

Primary Trig Derivatives

ddx[sin⁡x]=cos⁡xddx[cos⁡x]=−sin⁡x\frac{d}{dx}[\sin x] = \cos x \qquad \frac{d}{dx}[\cos x] = -\sin x
ddx[tan⁡x]=sec⁡2xddx[cot⁡x]=−csc⁡2x\frac{d}{dx}[\tan x] = \sec^2 x \qquad \frac{d}{dx}[\cot x] = -\csc^2 x
ddx[sec⁡x]=sec⁡xtan⁡xddx[csc⁡x]=−csc⁡xcot⁡x\frac{d}{dx}[\sec x] = \sec x\tan x \qquad \frac{d}{dx}[\csc x] = -\csc x\cot x

Key Derivatives (also from this unit)

ddx[ex]=exddx[ln⁡x]=1x\frac{d}{dx}[e^x] = e^x \qquad \frac{d}{dx}[\ln x] = \frac{1}{x}
AP Tip: Memory trick: 'co-' functions (cos, cot, csc) always get a negative sign. sec and csc derivatives bring back both functions.
Type 1

Basic Trig Derivatives

Apply the six trig derivative rules directly, combined with sum/difference and constant multiple rules.

Example 1
Find the derivative.
f(x)=3sin⁡x−4cos⁡x+2exf(x) = 3\sin x - 4\cos x + 2e^x
Example 2
Find the derivative.
y=2sec⁡x−3csc⁡x+cot⁡xy = 2\sec x - 3\csc x + \cot x

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Type 2

Product Rule with Trig

Combine the product rule with trig derivatives when the function is a product of a power and a trig function.

Example 3
Find the derivative.
ddx[x4cos⁡x]\frac{d}{dx}[x^4 \cos x]

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Type 3

Derivative from the Limit Definition

Use the limit definition f′(a) = lim[f(a+h)−f(a)]/h to find a specific derivative value.

Example 4
Use the limit definition to find f′(2).
f(x)=x2−3x+1f(x) = x^2 - 3x + 1

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