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HW $11$

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ess Calculus

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Shannon Sullivan

Question $16,4\mathrm{.}4\mathrm{.}47$

Part $1$ of $3$

HW Score: $02\mathrm{.}72\%,10\mathrm{.}04$ ot $16$ points

Points: $0\mathrm{.}33$ of $1$

Prove that if $y=y_0{e}^{kt},$ where $y_0$ and k are constants, then $\frac{dy}{dt}=ky.($This says that for exponential growth and decay, the rate of change of the population is proportional to the size of the population and the constant of proportionality is the growth or decay constant.$)$

The derivative of $(e^{(x)})$ equals $(e^{g(x)})g^{\text{'}}(x).$ Use this property to determine the derivative of $y_0{e}^{kt}.$

$\frac{d}{dt}(y_0{e}^{kt})=y_0[\frac{d}{dt}(e^{kt})]$

$=y_0$

$($Type an exact answer in terms of e$.)$