finding the derivative of a function using f'(x)=lna*a^x

Question

So the question states an animal population is given by P(t)=300(1.044)^t where t is the number of years since the study of the population began. find P'(5) and interpret your results.

 

So for the interpretation, I know what ever value I get, it would be five years after the study began the population increased by the rate     animals per year.

 

But I only know the interpretation not how to find p'(5). My attempt got as far as

P'(t)=300 limh→0 [(ln(1.044)(1.044^(t+h)-(300(1.044^(t+h))/h]

before it just fell apart. Could you please help me and walk me through this step by step because I am really confused. Thank you in advance.

Answer 1

Try using the logarithmic derivative.

 

y = Ba^t

 

(B, a are constants)

 

Take ln() on both sides:

 

ln(y) = lnB + t ln(a)

 

Take d/dt on both sides:

 

(1/y) dy/dt = ln(a)

 

dy/dt = ln(a) y = ln(a) Ba^t

 

Apply this to get the derivative of your function 

 

P(t) = 300 (1.044)^t

 

with B = 300, a = 1.044

 

Feel free to post a comment if you need more help.