# Help Please! A radioactive substance decaying so that after t years, the amount remaining, expressed as a percent of the original amount is a(t) = 100(1.1)^-t. determine the function, a', which represents the rate of decay of the substance.?

Apr 21, 2018

$a ' \left(t\right) = - 100 \left(\ln 1.1\right) {\left(1.1\right)}^{-} t$

#### Explanation:

$a ' \left(t\right)$, the rate of decay, can be found by differentiating $a \left(t\right) .$

$a \left(t\right) = 100 {\left(1.1\right)}^{-} t$

To make differentiation easier, rewrite using the fact that $x = {e}^{\ln} x :$

$a \left(t\right) = 100 {e}^{\ln \left[{\left(1.1\right)}^{-} t\right)}$

Recall that $\ln \left({a}^{x}\right) = x \ln a :$

$a \left(t\right) = 100 {e}^{- t \ln 1.1}$

We can now differentiate this using the Chain Rule and the rules for differentiating the exponential function of base $e .$

$a ' \left(t\right) = 100 {e}^{- t \ln 1.1} \cdot \frac{d}{\mathrm{dt}} - t \ln 1.1$

$a ' \left(t\right) = - 100 \left(\ln 1.1\right) {e}^{- t \ln 1.1}$

Recalling that ${e}^{- t \ln 1.1} = {e}^{\ln {\left(1.1\right)}^{-} t} = {\left(1.1\right)}^{-} t$, we get

$a ' \left(t\right) = - 100 \left(\ln 1.1\right) {\left(1.1\right)}^{-} t$