Derivatives of Natural Exponents

y = e^x is the inverse of y = ln x, so e^{ln x} = x and ln e^x = x  (using the property of inverse functions:  f(f ^-1(x)) = x and f ^-1(f(x)) = x.

Solve for x:

•            y = e ^x+1

      ln y = ln e ^x+1
ln y - 1 = x
 
• ln (2x - 3) = 5

             e⁵ = 2x - 3
       e⁵ + 3 = 2x
       e⁵ + 3 = x
          2

Properties of Exponents:

• e^ae^b = e^a+b
• e^a =  e^a-b

e^b
• (e^a)^b = e^ab

Graph y = e^x in your calculator.

Derivative of y = e^x:

g'(x) =       1
            f '(g(x))

        =       1
            f '(e^x)

       =       1
             1/e^x

       =   e^x

So e^x is its own derivative (natural, again)

Example:

      y' = x²e^x + e^x(2x)    (product rule)
         = x²e^x + 2xe^x
                    or xe^x(x² + 2)