The Derivative of the Exponential

Derivation of the Derivative

Before we derive the derivative, we need a few preliminary remarks.  Recall that the number e is defined by the limit as x approaches 0 of

f(x)  =  (1 + x)1/x

If we substitute Dx for x and let Dx be small then

(1 + Dx)1/Dx   @  e

Taking both sides to the Dx power we get

(1 + Dx)   @  eDx

Now its time to find the derivative of the exponential function f(x)  =  ex

The definition tells us

We have shown the following theorem

 Theorem (The derivative of ex) If                  f (x)  =  ex   then                f '(x)  =  f(x)  =  ex

Examples

Find the derivative of

1. e2x

2. x ex

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:

(x ex)'  =  (x)' (ex) + x (ex)'

=  ex + x ex

Exercises:

Find the derivatives of

1. 3 e4x

2.     ex

x
2

Application

A cross-section of the bottom half of an eggshell can be modeled by the equation

y  =  ex/4 + e-x/4         -4  <  x  <  4

where x is the horizontal measurement and y is the vertical measurement both measured in centimeters.  What is the vertical distance from the bottom of the shell to the middle of the shell?

Solution

First we need to find the bottom of the shell.  This will be where the shell obtains a minimum.  To find a minimum, we take a derivative and set it equal to zero.  We use the chain rule to get

y'  =  1/4 ex/4  -  1/4 e-x/4   =  0

ex/4 - e-x/4   =  0        Multiplying by 4

ex/4  =  e-x/4

x/4  =  -x/4

x  =  0

Plugging in gives

y(0)  =  e0/4 + e-0/4   =  1 + 1  =  2

To ensure that this is where the minimum is we take a second derivative

y''  =  1/16 ex/4  +  1/16 e-x/4

Plugging in 2 for x gives

y''(2)  =  1/16 e2/4  +  1/16 e-2/4

Which is positive (since exponentials are always positive).

Since the second derivative is positive the point is a minimum.

Now plug in the endpoints to get

y(-4)  =  e-4/4 + e4/4   @  3.1

y(4)  =  e4/4 + e-4/4   @  3.1

The vertical distance is

y(4) - y(0)   =  3.1 - 2  =  1.1

The bottom half of the eggshell is 1.1 centimeters high.

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