Differentiating Exponential Functions with Base e

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If y=e^x, dy/dx=e^x (Implicit Differentiation Proof)

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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^h-1}/{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^h-e^x}/h#
    by factoring out #e^x#,
    #=lim_{h to 0}{e^x(e^h-1)}/h=e^x lim_{h to 0}{e^h-1}/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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