I have seen around that the derivative of exponential functions like a^(x) is the function it self a^(x) times the derivative of the exponent (in this case is just 1) times ln(a).

Is this correct for any kind of exponential function?

Question

I have seen around that the derivative of exponential functions like a^(x) is the function it self a^(x) times the derivative of the exponent (in this case is just 1) times ln(a). 

Is this correct for any kind of exponential function?

phi · Answer

the idea is you can write a  as
$$a= e^{some\_number} $$
by definition, the some number is ln(a)
that is
$$ a= e^{\ln a} $$
thus
$$ a6x = \left( e^{\ln a}\right)^x = e^{x \ln a}$$

phi · Answer

*$a^x$

phi · Answer

ln a is a constant (call it c),  and the derivative of exp( c x) is exp( c x) * d (c x) or
c exp(c x)

anonymous · Answer

So, at the end we apply the definition of the derivative of "exp()", which actually I am not aware of, but apparently the derivative of "exp(x)" is exp(x) * (x)', right? But I am not seeing the connection between the function exp and e, are they the same?

phi · Answer

sorry,  yes exp(x)  means e^x  (just easier to type)