If P dollars is invested at an annual interest rate of r %, then at t years this investment grows to F dollars, where $F = P \left( 1 + \frac { r } { 100 } \right) ^ { t }$ .Find $\frac { d F } { d t }$ .
A) $P t \left( 1 + \frac { r } { 100 } \right) ^ { t - 1 }$
B) $\frac { P t } { 100 } \left( 1 + \frac { r } { 100 } \right) ^ { t - 1 }$
C) $\left( 1 + \frac { r } { 100 } \right) ^ { t }$
D) $\frac { P } { 100 } \ln \left( 1 + \frac { r } { 100 } \right) \left( 1 + \frac { r } { 100 } \right) ^ { t }$

Question

Quizplus · Accepted Answer

To find $\frac { d F } { d t }$, we apply the chain rule to the given function $F = P \left( 1 + \frac { r } { 100 } \right) ^ { t }$. The derivative of $F$ with respect to $t$ is $\frac { P } { 100 } \ln \left( 1 + \frac { r } { 100 } \right) \left( 1 + \frac { r } { 100 } \right) ^ { t }$, which accounts for the exponential growth rate of the investment over time.

If P Dollars Is Invested at an Annual Interest Rate F=P(1+r100)tF = P \left( 1 + \frac { r } { 100 } \right) ^ { t }F=P(1+100r​)t

If P Dollars Is Invested at an Annual Interest Rate $F = P \left( 1 + \frac { r } { 100 } \right) ^ { t }$