# 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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