Inverse Trig Functions (Leaving Cert Mathematics): Flashcards

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Inverse Trig Functions
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Derivative of cos⁡−1(xa)\cos^{-1}\left(\frac{x}{a}\right)

−1a2−x2\frac{-1}{\sqrt{a^2 - x^2}}

Derivative of sin⁡−1(xa)\sin^{-1}\left(\frac{x}{a}\right)

1a2−x2\frac{1}{\sqrt{a^2 - x^2}}

Derivative of tan⁡−1(xa)\tan^{-1}\left(\frac{x}{a}\right)

aa2+x2\frac{a}{a^2 + x^2}

When to use chain rule with inverse trig derivatives

With complex expressions involving inverse trig functions

Inner function in sin⁡−1(1/x2)\sin^{-1}(1/x^2) example

v=1x2v = \frac{1}{x^2}

Outer function in sin⁡−1(1/x2)\sin^{-1}(1/x^2) example

u=sin⁡−1(v)u = \sin^{-1}(v)

Derivative of v=1x2v = \frac{1}{x^2}

v′=−2x3v' = -\frac{2}{x^3}

u′u' for u=sin⁡−1(v)u = \sin^{-1}(v)

11−v2\frac{1}{\sqrt{1-v^2}}

ddxsin⁡−1(1x2)\frac{d}{dx}\sin^{-1}\left(\frac{1}{x^2}\right)

−2xx4−1-\frac{2}{x\sqrt{x^4-1}}

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