Derivatives of the inverse trigonometric functions

Remark — A look ahead: derivatives of the inverse trigonometric functions (Year 5)

In Year 5 we shall see that the inverse trigonometric functions have very elegant derivatives, which are derived from the theorem on the derivative of the inverse function: $\frac{d}{dx}\arcsin x = \frac{1}{\sqrt{1-x^2}}, \quad \frac{d}{dx}\arccos x = -\frac{1}{\sqrt{1-x^2}}, \quad \frac{d}{dx}\arctan x = \frac{1}{1+x^2}.$ As an important consequence for integrals: $\int \frac{dx}{1+x^2} = \arctan x + C, \quad \int \frac{dx}{\sqrt{1-x^2}} = \arcsin x + C.$ So $\arctan$ and $\arcsin$ “answer” integrals that would otherwise look mysterious. They are at the heart of many trigonometric substitutions in analysis.

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Topics: Trigonometry
Concepts: Arccosine · Arcsine · Arctangent · Derivative · Integral
Functions: Arccosine · Arcsine · Arctangent · Inverse trigonometric functions