Differentiating Exponential Functions with Base e
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Key Questions

This is one of the favorite function to take the derivatives of.
#y'=e^x# If you wish to find this derivative by the limit definition, then here is how we find it. First, we have to know the following property of
#e# :
#lim_{h to 0}{e^h1}/{h}=1# .
(Note: This means that the slope of#y=e^x# at#x=0# is#1# .)By the limit definition of the derivative, we have
#y'=lim_{h to 0}{e^{x+h}e^x}/h =lim_{h to 0}{e^x cdot e^he^x}/h#
by factoring out#e^x# ,
#=lim_{h to 0}{e^x(e^h1)}/h=e^x lim_{h to 0}{e^h1}/h#
by the property of#e# mentioned above,
#=e^x cdot 1=e^x# Hence, the derivative of
#e^x# is itself. 
#y'=e^(1/x)/(x^2)# Explanation :
Using Chain Rule,
Suppose,
#y=e^f(x)# then,
#y'=e^f(x)*f'(x)# Similarly following for the
#y=e^(1/x)# #y'=e^(1/x)*(1/x)'# #y'=e^(1/x)*(1/x^2)# #y'=e^(1/x)/(x^2)# 
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