Differentiating Exponential Functions with Base e

Key Questions

  • How do you find the derivative of y=e^(1/x)?

    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)

    Gaurav · 2 · Jul 30 2014
  • How do you find the derivative of y=e^x?

    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.

    Wataru · · Aug 31 2014

Questions

Differentiating Exponential Functions