Here are three ways to see that the derivative is #0#:
The power rule and chain rule
#d/dx (u^5) = 5u^4 d/dx(u)#
In this case #u = e# is a constant, so we get:
#d/dx (e^5) = 5e^4 d/dx(e) = 5e^4*0 = 0#
Exponential function and chain rule
#d/dx(e^u) = e^u d/dx(u)#
In this case #u = 5# is a constant, so we get:
#d/dx(e^5) = e^5 d/dx(5) = e^5*0 = 0#
#e^5# is a constant
#e ~~ 2.7#, so #e^5 # is s a number close to #2.7^5#.
The derivative of that number (a constant) is #0#
#d/dx(e^5) = 0#
Additional note This is a lot like asking for the derivative of #2^5# which is clearly the same as the derivative of #32# which is #0#.
The constant #e# causes confusion until a student gets comfortable with the fact that #e# is just some number.
Asking about the derivative of #x^e# also causes confusion .
