5.4 Exponential Functions: Differentiation and Integration

After this lesson, you should be able to: Develop properties of the natural exponential function. Differentiate natural exponential functions. Integrate natural exponential functions.

Definition of the Natural Exponential Function and Figure 5.19 The Natural Exponential Functions

Theorem 5.10 Operations with Exponential Functions

Properties of the Natural Exponential Function

Let’s consider the derivative of the natural exponential function. Difficult Way Going back to our limit definition of the derivative: First rewrite the exponential using exponent rules. Next, factor out e x. Since e x does not contain h, we can move it outside the limit. Derivatives of Natural Exponential Functions

Substituting h = 0 in the limit expression results in the indeterminate form, thus we will need to determine it. We can look at the graph of and observe what happens as x gets close to 0. We can also create a table of values close to either side of 0 and see what number we are closing in on. x f(x) Graph At x = 0, f (0) appears to be 1. Table As x approaches 0, f ( x ) approaches 1.

We can safely say that from the last slide that Thus Rule 1: Derivative of the Natural Exponential Function The derivative of the natural exponential function is itself.

Easy Way Going back to Differentiation of Inverse Function : Let, then. We have already known that So

Example 1 Find the derivative of f ( x ) = x 2 e x. Solution Do you remember the product rule? You will need it here. Product Rule: (1 st )(derivative of 2 nd ) + (2 nd )(derivative of 1 st ) Factor out the common factor xe x.

Example 2 Find the derivative of Solution We will need the chain rule for this one. Chain Rule: (derivative of the outside)(derivative of the inside)

Why don’t you try one: Find the derivative of. To find the solution you should use the quotient rule. Choose from the expressions below which is the correct use of the quotient rule.

No that’s not the right choice. Remember the Quotient Rule: (bottom)(derivative of top) – (top)(derivative of bottom) (bottom)² Try again. ReturnReturn

Good work! The quotient rule results in. Now simplify the derivative by factoring the numerator and canceling.

What if the exponent on e is a function of x and not just x ? Rule 2: If f ( x ) is a differentiable function then In words: the derivative of e to the f ( x ) is an exact copy of e to the f ( x ) times the derivative of f ( x ).

Theorem 5.11 Derivative of the Natural Exponential Function

Example 3 Find the derivative of Solution We will have to use Rule 2. The exponent, 3 x is a function of x whose derivative is 3. An exact copy of the exponential function Times the derivative of the exponent

Example 4 Find the derivative of Solution Again, we used Rule 2. So the derivative is the exponential function times the derivative of the exponent. Or rewritten:

Example 5 Differentiate the function SolutionUsing the quotient rule Keep in mind that the derivative of e -t is e -t (-1) or -e -t Recall that e 0 = 1. Distribute e t into the ( )’s

You try: Find the derivative of. Click on the button for the correct answer.

No, the other answer was correct. Remember when you are doing the derivative of e raised to the power f(x) the solution is e raised to the same power times the derivative of the exponent. What is the derivative of ? Try again. ReturnReturn

Good work!! Here is the derivative in detail.

Example 6 A quantity growing according to the law where Q 0 and k are positive constants and t belongs to the interval experiences exponential growth. Show that the rate of growth Q ’ ( t ) is directly proportional to the amount of the quantity present. Solution Remember: To say Q ’ ( t ) is directly proportional to Q ( t ) means that for some constant k, Q ’ ( t ) = kQ ( t ) which was easy to show.

Example 7 Find the inflection points of Solution We must use the 2 nd derivative to find inflection points. First derivative Product rule for second derivative Simplify Set equal to 0. Exponentials never equal 0. Set the other factor = 0. Solve by square root of both sides.

To show that they are inflection points we put them on a number line and do a test with the 2 nd derivative: Intervals Test Points Value 0 1 f ” (-1)= 4e -1 – 2e -1 =2e -1 > 0 f ” (0)=0 – 2 = –2 < 0 f ” (1)= 4e -1 – 2e -1 = 2e -1 > Since there is a sign change across the potential inflection points, and are inflection points.

In this lesson you learned two new rules of differentiation and used rules you have previously learned to find derivatives of exponential functions. The two rules you learned are: Rule 1: Derivative of the Natural Exponential Function Rule 2: If f ( x ) is a differentiable function then

Integrals of Natural Exponential Functions Each rules of differentiation has a corresponding integration rule. Rule 2 : If f ( x ) is a differentiable function then Rule 1 : Derivative of the Natural Exponential Function Rule 2 : If f ( x ) is a differentiable function then Rule 1 : Integral of the Natural Exponential Function

Theorem 5.12 Integration Rules for Exponential Functions

Example 8 Find Solution We must use Rule 2 of Integration. Make an f ( x ) or u in the “ d ” Apply the Rule 2 of Integration

Example 9 Find Solution We must use Rule 2 of Integration. Make an f ( x ) or u in the “ d ” Apply the Rule 2 of Integration

Example 10 Find Solution We must use Rule 2 of Integration. Try to make an f ( x ) or u in the “ d ” Apply the Rule 2 of Integration Made an f ( x ) or u in the “ d ”

Example 11 Find Solution We will use Integration Rule of Basic Trig of Functions. Use Rule 2 of Derivative of N. Exp. Func. Made an u in the “ d ” Apply the Integration Rule for Basic Trig Function

Example 12 Find Solution We must use Rule 2 of Integration. Use Rule 2 of Derivative of N. Exp. Func. Apply the Log Rule for Integration

Homework 5.4 P. 356 Q xxxxxxxxxxxx