The Derivative and Integral of the Exponential Function

Definitions and Properties of the Exponential Function

The exponential function,

y = ex

is defined as the inverse of

ln x

Therefore

ln(ex) = x

and

elnx = x

Recall that

1. eaeb  =  ea + b

2.  ea/eb  =  e(a - b)

Proof of 2.

ln[ ea/eb]  =  ln[ea] - ln[eb

= a - b  =  ln[e a - b]

since  ln(xis 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/ex

Hence

g'(x)  =  ex

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

Examples:

Find the derivative of

1. e2x

2. xex

Solution

1. We use the chain rule with

y = eu,     u = 2x

Which gives

y' = eu,     u' = 2

So that

(e2x)' = (eu)(2) = 2e2x

2. We use the product rule:

(xex)' = (x)' (ex) + x (ex)'

= ex + x ex

Exercises:

Find the derivatives of

1. ln(ex) 2. ex /x2 Examples

A. B. Solution

1. Since

ex   =  (ex)'

We can integrate both sides to get ex dx  =  ex + C

2. For this integral, we can use u substitution with

u = ex,      du = ex dx

The integrals becomes = eu + C Exercises

Integrate:

1.  2.  