Question

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

Answer 1

You can answer the question by calculating the first derivative. First of all, applying the rule `a^(-x)=1/(a^x)`, we have that `(1/3)^{-x}=3^x`. 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.