Question

Determine the instantaneous rate of change at x=2 using the functions:a) f(x)= 2/xb) g(x)= 3^x

Answer 1

Instantaneous rate of change can be found by finding the derivative of the equation.

a) Given the function

`f(x)=(2)/(x)`

The equation in this particular problem can be rewritten as follows:

`f(x)=2(x^(-1))`

The derivate is

`f^(\prime)(x)=-2(x^(-1-1))=-2(x^(-2))=-(2)/(x^2)`

From here we can plug in our given, x=2, and get the answer

`f^(\prime)(2)=-(2)/((2)^2)=-(2)/(4)=-(1)/(2)`

Answer: -1/2

b) The function

`g(x)=3^x`

Applying the power rule for derivatives

`g^(\prime)(x)=3^x\ln (3)`

And we can plug in our given, x=2, and get the answer

`g^(\prime)(2)=3^2\ln (3)=9\ln (3)=9.89`

Answer: 9.89