Instantaneous rate of change can be found by finding the derivative of the equation.
a) Given the function
[tex]f(x)=\frac{2}{x}[/tex]The equation in this particular problem can be rewritten as follows:
[tex]f(x)=2(x^{-1})[/tex]The derivate is
[tex]f^{\prime}(x)=-2(x^{-1-1})=-2(x^{-2})=-\frac{2}{x^2}[/tex]From here we can plug in our given, x=2, and get the answer
[tex]f^{\prime}(2)=-\frac{2}{(2)^2}=-\frac{2}{4}=-\frac{1}{2}[/tex]Answer: -1/2
b) The function
[tex]g(x)=3^x[/tex]Applying the power rule for derivatives
[tex]g^{\prime}(x)=3^x\ln (3)[/tex]And we can plug in our given, x=2, and get the answer
[tex]g^{\prime}(2)=3^2\ln (3)=9\ln (3)=9.89[/tex]Answer: 9.89
