AP Calculus AB — calculusStandard1 min read

Trigonometric Derivatives

Differentiate all six trig functions for AP Calculus AB.

Tutor AI Team
Tutor AI Team

Knowing derivatives of all six trig functions is essential for AP Calculus AB.

Derivatives

f(x)f(x) f′(x)f'(x)
sin⁡x\sin x cos⁡x\cos x
cos⁡x\cos x −sin⁡x-\sin x
tan⁡x\tan x sec⁡2x\sec^2 x
csc⁡x\csc x −csc⁡xcot⁡x-\csc x \cot x
sec⁡x\sec x sec⁡xtan⁡x\sec x \tan x
cot⁡x\cot x −csc⁡2x-\csc^2 x

With Chain Rule

ddx[sin⁡(u)]=cos⁡(u)⋅u′\frac{d}{dx}[\sin(u)] = \cos(u) \cdot u'

ddx[tan⁡(3x)]=3sec⁡2(3x)\frac{d}{dx}[\tan(3x)] = 3\sec^2(3x)

Worked Examples

y=sin⁡(x2)y = \sin(x^2) → y′=2xcos⁡(x2)y' = 2x\cos(x^2).

y=sec⁡(5x)y = \sec(5x) → y′=5sec⁡(5x)tan⁡(5x)y' = 5\sec(5x)\tan(5x).

Practice Problems

    1. ddx[cos⁡(3x2)]\frac{d}{dx}[\cos(3x^2)].
    1. ddx[xtan⁡x]\frac{d}{dx}[x\tan x] (product rule).
    1. ddx[csc⁡(ex)]\frac{d}{dx}[\csc(e^x)].

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Key Takeaways

  • ✓

    Memorize all 6 derivatives.

  • ✓

    Apply chain rule for composite trig functions.

  • ✓

    'Co-' functions have negative derivatives.

Tags:
#ap#calculus#differentiation#trigonometry

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