Is $\displaystyle 3^z$ a constant or a variable?
Hopefully others can supplement the following with more precise language...
3^(something) could be interpreted in a few ways, depending on what that "something" is.
If "something" is a number, then it's obvious: 3^2 = 3^2(!) = 9.
If "something" is, say, a/b/c/d, then we might interpret a/b/c/d to be a constant, in which case 3^a, while still "depending" on the value of a, doesn't vary once we have defined a.
If "something" is, say, u/x/y/z, then - as everyone has to this point - we interpret these letters are being variable.
Thoughts?
See http://en.wikipedia.org/wiki/Variable_%28mathematics%29 (under heading "Notation").
Yeah, I know. It's wikipedia. But this way, we're all pulling from the same definition!
We will assume that z is a variable (by convention - see my previous post).
The derivative of $\displaystyle a^z$ is $\displaystyle a^z*ln(a)*z'$
This is an exponential function with base 3. Notice that we use the chain rule (hence the z' in the derivative) and when the base is "e", then you just get
$\displaystyle e^z*ln(e)z' = $
$\displaystyle e^z*1*z' =$
$\displaystyle e^zz'$