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All derivatives of circular trigonometric functions can be found from those of sin(x) and cos(x) by means of the quotient rule applied to functions such as tan(x) = sin(x)/cos(x). Knowing these derivatives, the derivatives of the inverse trigonometric functions are found using implicit differentiation.
For arcsine, the series can be derived by expanding its derivative, , as a binomial series, and integrating term by term (using the integral definition as above). The series for arctangent can similarly be derived by expanding its derivative 1 1 + z 2 {\textstyle {\frac {1}{1+z^{2}}}} in a geometric series , and applying the integral definition ...
In mathematics, the arctangent series, traditionally called Gregory's series, ... The derivative of arctan x is 1 / (1 + x 2); conversely, the integral of 1 / ...
The derivatives in the table above are for when the range of the inverse secant is [,] and when the range of the inverse cosecant is [,]. It is common to additionally define an inverse tangent function with two arguments , arctan ( y , x ) . {\displaystyle \arctan(y,x).}
3 Arctangent function integration formulas. ... Differentiation of trigonometric functions – Mathematical process of finding the derivative of a trigonometric function;
A ray through the unit hyperbola = in the point (,), where is twice the area between the ray, the hyperbola, and the -axis. The earliest and most widely adopted symbols use the prefix arc-(that is: arcsinh, arccosh, arctanh, arcsech, arccsch, arccoth), by analogy with the inverse circular functions (arcsin, etc.).
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As the function atan2 is a function of two variables, it has two partial derivatives. At points where these derivatives exist, atan2 is, except for a constant, equal to arctan(y/x). Hence for x > 0 or y ≠ 0,