Derivative of Exponential Help
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,
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
Compute the derivatives:
For the first problem, notice that u = 4 x hence du/dx = 4. Therefore we have
You Try It : Calculate ( d/dx )(exp( x · sin x )).
Calculate the integrals:
Evaluate the integral
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
Evaluate the integral
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.
Simplify the expression
ln( e 5 · 8 −3).
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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