Rate of Exponential Growth

Theorem

Let $y = a e^{b t}$ be an exponential growth function.

Then the rate of growth of $y$ is proportional to the value of $y$ such that:

$\dfrac {\d y} {\d t} = b y$

Proof

\(\ds y\)

\(=\)

\(\ds a e^{b t}\)

\(\ds \leadsto \ \ \)

\(\ds \dfrac {\d y} {\d t}\)

\(=\)

\(\ds a b e^{b t}\)

Derivative of Exponential Function: Corollary $1$

\(\ds \)

\(=\)

\(\ds b y\)

Definition of $y$

$\blacksquare$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): exponential growth or decay
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): exponential growth or decay