Derivative of Exponential Functions with Other Logs

[V0ODO0CH1LD]

Can the derivative of an exponential function be calculated with logs base something other than e? Like base 10 or 2?

 

[micromass]

An example of what you're trying to do would be welcome.

 

[theorem4.5.9]

You can, but you will get an extra constant from the chain rule. Just use a change of base formula for log or rewrite an exponential in some other base and the term comes right out.

$e$ is important precisely because it's the only base of the exponential where after differentiating the constant is $1$.

 

[V0ODO0CH1LD]

You can, but you will get an extra constant from the chain rule. Just use a change of base formula for log or rewrite an exponential in some other base and the term comes right out.

$e$ is important precisely because it's the only base of the exponential where after differentiating the constant is $1$.

Could you maybe expand on that? Or point out somewhere where I could read about it?

If I start with "f(x+Δx) - f(x) = a^(x+Δx) - a^x" how do I continue with log_2 or anything else from here? Would I even go to that being equal to "a^x * (a^Δx - 1)"?