Differentiate It Away

Calculus Level 3

Suppose we have the following relationship:

\(a(t) = b*c(t) + e*\frac{d}{dt} c(t)\)

The "\(*\)" above denotes multiplication. As shown above, \(a(t)\) and \(c(t)\) are functions of the parameter \(t\), and parameters \(b\) and \(e\) are constants. Suppose that we want to solve for \(e\) without referring to \(b\). The parameter \(e\) can be written as:

\( \Large{e=\frac{c(t) * \frac{d^{W}}{dt^{W}} a(t) -a(t) * \frac{d^{X}}{dt^{X}} c(t) }{c(t) * \frac{d^{Y}}{dt^{Y}} c(t) -(\frac{d^{Z}}{dt^{Z}} c(t))^{2} }}\)

Parameters \(W, X, Y, \) and \(Z\) denote the numbers of derivatives of their respective functions. Determine \((W+X+Y+Z\)).


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