The equation $A(t)=2000e^{0\mathrm{.}05t}$ gives the balance after $t$ years of an initial investment of $2000$ dollars which pays $5\mathrm{.}00\%$ compounded continuously.

a$.$ Find a formula for $\frac{dA}{dt}$

b$.$ Find and interpret $A^{\text{'}}(9).$ Include appropriate units.

c$.$ Compare the approximation of $$157$ to the actual change. Report your answer to two decimal places.

$5\mathrm{.}00\%$

B$.$ The future value of a $10$ year investment of $$2000$ will be $$157$ more than the future value of a $9$ year investment of $$2000.$

C$.$ The future value of a $10$ year investment of $$2001$ will be $ more than the future value of a $9$ year investment of $$2000.$

D$.$ The future value of a $9$ year investment of $$2001$ will be $ $$ more than the future value of a $9$ year investment of $$2000.$

c$.A(10)-A(9)=$ dollars per year. $($Round to two decimal places as needed.$)$