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acos_forward¶
View page sourceInverse Cosine and Hyperbolic Cosine Forward Mode Theory¶
Derivatives¶
If \(F(x)\) is \(\R{acos} (x)\) or \(\R{acosh} (x)\) the corresponding derivative satisfies the equation
and in the standard math function differential equation , \(A(x) = 0\), \(B(x) = \sqrt{ \mp( x * x - 1 ) }\), and \(D(x) = \mp 1\). We use \(a\), \(b\), \(d\) and \(z\) to denote the Taylor coefficients for \(A [ X (t) ]\), \(B [ X (t) ]\), \(D [ X (t) ]\), and \(F [ X(t) ]\) respectively.
Taylor Coefficients Recursion¶
We define \(Q(x) = \mp ( x * x - 1 )\) and let \(q\) be the corresponding Taylor coefficients for \(Q[ X(t) ]\). It follows that
It follows that \(B[ X(t) ] = \sqrt{ Q[ X(t) ] }\) and from the equations for the square root that for \(j = 0 , 1, \ldots\),
It now follows from the general Taylor Coefficients Recursion Formula that for \(j = 0 , 1, \ldots\),