# 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 \((W+X+Y+Z\)).