Question

Ya'll fr get 50 points if you answer this.
What is the rate of change of
`f(x)=2^x`?

Answer 1

`(dy)/(dx)=2^x\ln 2`

Explanation:

**This is a non-linear function and therefore does not have a constant rate of change. It will have a different slope depending on what points you use in the average rate of change formula:
`\mathsf{average \ rate \ of \ change = (change \ in \ y)/(change \ in \ x)}`

To calculate rate of change, differentiate.

substitute y for
`f(x)`:

`\implies y=2^x`

Take natural logs of both sides:

`\implies \ln y=\ln 2^x`

Apply the log rule
`\ln a^b=b \ln a` :

`\implies \ln y=x\ln 2`

Differentiate with respect to
`x`:

`\implies (1)/(y) (dy)/(dx)=\ln 2`

Mulitply both sides by
`y`:

`\implies (dy)/(dx)=y\ln 2`

Replace
`y` with
`y=2^x`

`\implies (dy)/(dx)=2^x\ln 2`

Therefore, rate of change of the function is :

`(dy)/(dx)=2^x\ln 2`