Is the function y = -5(1/3)^ -xy=5(13)x exponential growth or decay?

1 Answer
Mar 29, 2015

You can answer the question by calculating the first derivative. First of all, applying the rule a^(-x)=1/(a^x)ax=1ax, we have that (1/3)^{-x}=3^x(13)x=3x. Then, since you can factor out constants, you have that
d/dx -5(1/3)^{-x}= -5\ d/dx (1/3)^{-x} = -5\ d/dx 3^x
As a fundamental derivative, we know that d/dx 3^x=3^x*log(3).

So, the first derivative is -5\log(3)*3^x, which is always negative since 5\log(3)*e^x is always positive. So, your function is always decreasing, and you have exponential decay.