The Derivative and Integral of the Exponential Function

Definitions and Properties of the Exponential Function

The exponential function, 

        y = e^x  

is defined as the inverse of 

        ln x  

Therefore

        ln(e^x) = x 

and 

        e^lnx = x

Recall that

1. e^ae^b  =  e^{a + b}  
   

2.  e^a/e^b  =  e^{(a - b)}
   

Proof of 2.  

        ln[ e^a/e^b]  =  ln[e^a] - ln[e^b] 

        = a - b  =  ln[e ^{a - b}]

    since  ln(x) is 1-1, the property is proven.

The Derivative of the Exponential

    
We will use the derivative of the inverse theorem to find the derivative of the exponential.  The derivative of the inverse theorem says that if f and g are inverses, then

                            1
        g'(x)  =                     
                        f '(g(x))

Let 

        f (x) = ln(x) 

then

        f '(x) = 1/x 

so that

        f '(g(x)) = 1/e^x  

Hence

        g'(x)  =  e^x

       

Theorem If             f (x) = ex   then           f '(x) = f (x) = ex

Examples:

Find the derivative of

1. e^2x
   

2. xe^x

Solution

1. We use the chain rule with
        
           y = e^u,     u = 2x
   
   Which gives
   
            y' = e^u,     u' = 2
   
   So that
   
           (e^2x)' = (e^u)(2) = 2e^2x 
   

2. We use the product rule:
   
           (xe^x)' = (x)' (e^x) + x (e^x)' 
   
           = e^x + x e^x 
   
   

Exercises:

Find the derivatives of 

1. ln(e^x)       
   

2. e^x /x²       

Examples

A. 

B. 

Solution

1. Since
   
            e^x   =  (e^x)'
   
   We can integrate both sides to get
   
           e^x dx  =  e^x + C
   

2. For this integral, we can use u substitution with
   
           u = e^x,      du = e^x dx
   
   The integrals becomes

          
   
           = e^u + C  
   
           
   
   

Exercises

Integrate:

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