Respuesta :
Remember the following properties of the derivatives:
Derivative of an exponential function:
[tex]\frac{d}{dx}ke^x=ke^x[/tex]Chain rule:
[tex]\frac{d}{dx}f(u)=\frac{d}{du}f(u)\cdot\frac{du}{dx}[/tex]Also, remember the following property of exponentials:
[tex]a^x=e^{x\cdot\ln (a)}[/tex]Use these properties to find the derivative of the following function:
[tex]f(x)=k\cdot a^x[/tex]The function can be rewritten as:
[tex]f(x)=k\cdot e^{x\cdot\ln (a)}[/tex]Let:
[tex]u=x\cdot\ln (a)[/tex]Then:
[tex]\begin{gathered} \frac{d}{dx}k\cdot e^{x\cdot\ln (a)}=\frac{d}{dx}k\cdot e^u \\ =\frac{d}{du}(k\cdot e^u)\cdot\frac{du}{dx} \\ =k\cdot e^u\cdot\frac{d}{dx}(x\cdot\ln (a)) \\ =k\cdot e^{x\cdot\ln (a)}\cdot\ln (a) \\ =k\cdot a^x\cdot\ln (a) \end{gathered}[/tex]In this case, we can see that k=20 and a=1.31:
[tex]y=20\cdot1.31^x[/tex]Then, the derivative of this function, is:
[tex]y^{\prime}=20\cdot1.31^x\cdot\ln (1.31)[/tex]Therefore, the rate of increase of the given function, is:
[tex]y^{\prime}=20\cdot\ln (1.31)\cdot(1.31)^x[/tex]