Inverse Trigonometric Functions - Derivatives of Inverse Trigonometric Functions

Derivatives of Inverse Trigonometric Functions

Simple derivatives for real and complex values of x are as follows:

\begin{align} \frac{d}{dx} \arcsin x & {}= \frac{1}{\sqrt{1-x^2}}\\ \frac{d}{dx} \arccos x & {}= \frac{-1}{\sqrt{1-x^2}}\\ \frac{d}{dx} \arctan x & {}= \frac{1}{1+x^2}\\ \frac{d}{dx} \arccot x & {}= \frac{-1}{1+x^2}\\ \frac{d}{dx} \arcsec x & {}= \frac{1}{x\,\sqrt{x^2-1}}\\ \frac{d}{dx} \arccsc x & {}= \frac{-1}{x\,\sqrt{x^2-1}} \end{align}

Only for real values of x:

\begin{align} \frac{d}{dx} \arcsec x & {}= \frac{1}{|x|\,\sqrt{x^2-1}}; \qquad |x| > 1\\ \frac{d}{dx} \arccsc x & {}= \frac{-1}{|x|\,\sqrt{x^2-1}}; \qquad |x| > 1 \end{align}

For a sample derivation: if, we get:

Read more about this topic:  Inverse Trigonometric Functions

Famous quotes containing the words inverse and/or functions:

    “The quality of moral behaviour varies in inverse ratio to the number of human beings involved.”
    —Aldous Huxley (1894–1963)

    “Those things which now most engage the attention of men, as politics and the daily routine, are, it is true, vital functions of human society, but should be unconsciously performed, like the corresponding functions of the physical body.”
    —Henry David Thoreau (1817–1862)