**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**

**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* )).

**Examples**

**Example 2**

Calculate the integrals:

**Solution 2**

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*

*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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