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

Question

Quizplus · Accepted Answer

To find $\frac{dF}{dt}$, we apply the derivative with respect to $t$ to the formula $F = P \left( 1 + \frac { r } { 100 } \right) ^ { t }$. Using the chain rule, the derivative of $a^t$ (where $a$ is a constant) is $a^t \ln(a)$. Thus, $\frac{dF}{dt} = P \left( 1 + \frac { r } { 100 } \right) ^ { t } \ln \left( 1 + \frac { r } { 100 } \right)$.

If P Dollars Are Invested at an Annual Rate of R