which derivative is described by the following equation lim h approaches 0 of ((2^(3+h) - 2^3))/h

Question

limit as h approaches 0 of 2 ^ (3+h) -2^3 all over h

 

a) g prime (3) where g(x) = 2^(3+x)

 

b) g prime (3) where g(x) = 2^x

Answer 1

By the definition of derivative

 

lim        g(x+h) - g(x)

h->0    ______________

                    h

 

 

If g(x)=2^x  , then  we would have

 

lim     2^{(x + h)} - 2^x

h->0  ____________

                   h

 

 

Plugging x=3, we then have

 

lim      2^{(3 + h)} - 2³

h->0  __________

                 h

 

 

This is the same as the derivative of g(x) when x=3.

 

ANS is B.

Answer 2

The answer is b), since the function in question is 2^x.