Find the relative rate of change of f (x) = 12+2e^-2x

The relative rate of change of a function f (x) is the ratio of the derivative f' (x) to the function f (x). Since f' (x) = - 4e^ (-2x) the relative rate of change r = [-4e^ (-2x) ]/[12 + 2e^ (-2x) ]. This can be simplified by factoring a value of two resulting in: r = [-2e^ (-2x) ]/[6 + e^ (-2x) ]. This can be further re-arranged if desired into: r = - 1/3 e^ (-2x) - 2

The rate of change of a given function is given by f' (x) = dy/dx

given that the function is:

f (x) = 12+2^ (-2x)

f' (x) = 2 * (-2) e^ (-2x)

f' (x) = - 4e^ (-2x)

the relative rate of change will be the ratio of the derivative to the original function:

f' (x) / f (x)

= (12+2^ (-2x)) / (-4e^ (-2x))