Question

the growth of a bacteria each hour is given by the function f(x)= 5500(1.65)^x. At what percent are the bacteria growing each hour?

Answer 1

The growth function per hour is
f(x) = 5500(1.65)ˣ

To find the growth rate per hour, we should find the derivative of f(x).
Take the natural log of the equation.
ln(f) = ln[5500(1.65)ˣ]
      = ln(5500) + x ln(1.65)

Take the derivative with respect to x.
f'/f = ln(1.65) = 0.5
Therefore the derivative is
f' = 0.5f
   = 2750(1.65)ˣ
   = 275000(1.65)ˣ percent or 2.75 x 10⁵ (1.65)ˣ percent

Answer:
$f^{'} \equiv \frac{df}{dx}=2.75\times10^{5}(1.65)^{x}\,percent$