\(\newcommand{\W}[1]{ \; #1 \; }\) \(\newcommand{\R}[1]{ {\rm #1} }\) \(\newcommand{\B}[1]{ {\bf #1} }\) \(\newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} }\) \(\newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} }\) \(\newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} }\) \(\newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }\)
pow_forward
Power Function Forward Mode Theory
We consider the operation \(F(x) = x^y\) where \(x\) is a variable and \(y\) is a parameter.
Derivatives
The corresponding derivative satisfies the equation
This is the standard math function differential equation , where \(A(x) = y\), \(B(x) = x\), and \(D(x) = 0\). 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. It follows that \(b^j = x^j\), \(d^j = 0\),
Taylor Coefficients Recursion
z^(0)
e^(j)
z^j
For \(j = 0, \ldots , p-1\)
For \(j = 1, \ldots , p\)