The Derivative and Integral of the Exponential Function

I.  Quiz

II.  Homework

III.  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.

IV.  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

    g'(x) = 1/f'(g(x))

    Let f(x) = ln(x) then

    f'(x) = 1/x so that

    f'(g(x)) = 1/e^x  Hence

    g'(x) = 1/1/e^x  = e^x

Theorem:

    If f(x) = e^x then

    f'(x) = f(x) = e^x

Examples:

    Find the derivative of

1. e^2x

2. xe^x

3. ln(e^x)

4. e^x /x²

Integrate:

1. int e^x dx

2. int xe^x2dx

3. int (cosx)e^sinx dx