# A bacteria culture initially contains 100 cells and grows at

A bacteria culture initially contains 100 cells and grows at a rate proportional to its size. After an hour the population has increased to 420. Find the rate of growth after 3 hours.
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Marcus Herman

From part(a), we know that
$P\left(t\right)=100\left(4.2{\right)}^{t}$
Differentiate

Recall that:

${P}^{\prime }\left(t\right)=100{\left(4.2\right)}^{t}\cdot \mathrm{ln}4.2$
${P}^{\prime }\left(t\right)=P\left(t\right)\cdot \mathrm{ln}4.2$
Substitute t=3
${P}^{\prime }\left(3\right)=P\left(3\right)\cdot \mathrm{ln}4.2$
From part (b), we know that $P\left(3\right)=7409$
${P}^{\prime }\left(3\right)=7409\cdot \mathrm{ln}4.2\approx 10633$ Bacteria/hour
Result: ${P}^{\prime }\left(3\right)=7409\cdot \mathrm{ln}4.2\approx 106333$ Bacteria/hour