Question

What is the derivative of `b^x` where b is a constant?

Answer 1

`d/dx[b^x]=b^x*lnb`

Explanation:

First, note that

`b^x=e^ln(b^x)=e^(xlnb)`

This allows us to differentiate the function using the chain rule:

`d/dx[e^(xlnb)]=e^(xlnb)*d/dx[xlnb]`

Just like `d/dx[5x]=5`, `d/dx[xlnb]=lnb`, since `lnb` will always be a constant.

This gives us a derivative of:

`e^(xlnb)*lnb`

Now, recall that `e^(xlnb)=b^x`. This gives us our final, differentiated result:

`d/dx[b^x]=b^x*lnb`