The value of an investment in Canada Savings Bonds is modelled by $A(t) = A_0 e^{0.0255t}$... Rest of question below. The value of an investment in Canada Savings Bonds is modeled by $$A(t) = A_0 e^{0.0255t}$$, where A is the amount the investment is worth after $t$ years, and $A_0$ is the initial amount invested. At what rate, correct to 3 decimal places, is the investment growing at the time when its value has doubled? I understand that we have to find the derivative but how can we find it if we don't know an initial amount?

The question asks for rate _when the investments value has been doubled_ or when

$$A(t)=2A_0 = A_0 e^{0.0255t}$$ So the initial value cancels and you have: $$ 2=e^{0.0255t}\Rightarrow \ln(2)=0.0255t\Rightarrow t=\frac{\ln(2)}{0.0255} $$ Now, taking the derivative: $$A'(t)=0.0255A_0e^{0.0255t}\Rightarrow A'(\frac{\ln(2)}{0.0255})=0.0255A_0e^{0.0255(\frac{\ln(2)}{0.0255})}\\\ =.0255A_0e^{\ln(2)}=2(.0255)A_0=.051A_0$$ Which does depend on the principal, as it should.