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# Derivative of Exponential Help

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By — McGraw-Hill Professional
Updated on Aug 31, 2011

## Calculus Properties of the Exponential

Now we want to learn some “calculus properties” of our new function exp( x ). These are derived from the standard formula for the derivative of an inverse, as in Section 2.5.1.

For all x we have

In other words,

More generally,

and

We note for the record that the exponential function is the only function (up to constant multiples) that is its own derivative. This fact will come up later in our applications of the exponential

#### Example 1

Compute the derivatives:

#### Solution 1

For the first problem, notice that u = 4 x hence du/dx = 4. Therefore we have

Similarly,

You Try It : Calculate ( d/dx )(exp( x · sin x )).

#### Example 2

Calculate the integrals:

We have

#### Example 3

Evaluate the integral

#### Solution 3

For clarity, we let φ( x ) = cos 3 x , φ′( x ) = 3 cos 2 x · (− sin x ). Then the integral becomes

Resubstituting the expression for φ( x ) then gives

#### Example 4

Evaluate the integral

#### Solution 4

For clarity, we set φ( x ) = exp( x ) − exp(− x ), φ′( x ) = exp( x ) + exp(− x ). Then our integral becomes

Resubstituting the expression for φ( x ) gives

You Try It : Calculate ∫ x · exp( x 2 − 3) dx.

## The Number e

The number exp(1) is a special constant which arises in many mathematical and physical contexts. It is denoted by the symbol e in honor of the Swiss mathematician Leonhard Euler (1707–1783) who first studied this constant. We next see how to calculate the decimal expansion for the number e .

In fact, as can be proved in a more advanced course, Euler’s constant e satisfies the identity

This formula tells us that, for large values of n, the expression

gives a good approximation to the value of e. Use your calculator or computer to check that the following calculations are correct:

With the use of a sufficiently large value of n, together with estimates for the error term

it can be determined that

e = 2.71828182846

to eleven place decimal accuracy. Like the number , the number e is an irrational number. Notice that, since exp(1) = e, we also know that ln e = 1.

#### Example

Simplify the expression

ln( e 5 · 8 −3).

#### Solution

We calculate that

You Try It : Use your calculator to compute log 10 e and ln 10 = log e 10. Confirm that these numbers are reciprocals of each other.

Find practice problems and solutions for these concepts at: Transcendental Functions Practice Test.

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#### Q:

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