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Derivative of Exponential Help

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

Transcendental Functions 6.2 Exponential Basics

In other words,

Transcendental Functions 6.2 Exponential Basics

More generally,

Transcendental Functions 6.2 Exponential Basics

and

Transcendental Functions 6.2 Exponential Basics

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

Example 1

Compute the derivatives:

Transcendental Functions 6.2 Exponential Basics

Solution 1

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

Transcendental Functions 6.2 Exponential Basics

Similarly,

Transcendental Functions 6.2 Exponential Basics

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

Examples

Example 2

Calculate the integrals:

Transcendental Functions 6.2 Exponential Basics

Solution 2

We have

Transcendental Functions 6.2 Exponential Basics

Example 3

Evaluate the integral

Transcendental Functions 6.2 Exponential Basics

Solution 3

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

Transcendental Functions 6.2 Exponential Basics

Resubstituting the expression for φ( x ) then gives

Transcendental Functions 6.2 Exponential Basics

Example 4

Evaluate the integral

Transcendental Functions 6.2 Exponential Basics

Solution 4

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

Transcendental Functions 6.2 Exponential Basics

Resubstituting the expression for φ( x ) gives

Transcendental Functions 6.2 Exponential Basics

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

Transcendental Functions 6.2 Exponential Basics

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

Transcendental Functions 6.2 Exponential Basics

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

Transcendental Functions 6.2 Exponential Basics

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

Transcendental Functions 6.2 Exponential Basics

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

Transcendental Functions 6.2 Exponential Basics

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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