# Arithmetic Course/Non Linear Function/Exponential function/Exponential growth

## Exponential Growth

Exponential Growth function is a function that has the value growth exponentially

${\displaystyle N(t)=N_{0}e^{\lambda t}.\,}$

Which can be proven as the root of a Differential Equation of the form

${\displaystyle {\frac {dN}{dt}}=\lambda N.}$

## Proof

For a Differential Equation of the form

${\displaystyle {\frac {dN}{dt}}=\lambda N.}$
${\displaystyle \int {\frac {dN}{N}}=\lambda \int dt}$
${\displaystyle lnN=\lambda t+c}$
${\displaystyle N=e^{(}\lambda t+c)}$
${\displaystyle N(t)=N_{0}e^{\lambda t}.\,}$