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5.4 Differentiation of Exponential Functions

5.4 Differentiation of Exponential Functions. By Dr. Julia Arnold and Ms. Karen Overman using Tan’s 5th edition Applied Calculus for the managerial , life, and social sciences text. Let’s consider the derivative of the exponential function.

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5.4 Differentiation of Exponential Functions

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  1. 5.4 Differentiation of Exponential Functions By Dr. Julia Arnold and Ms. Karen Overman using Tan’s 5th edition Applied Calculus for the managerial , life, and social sciences text

  2. Let’s consider the derivative of the exponential function. Going back to our limit definition of the derivative: First rewrite the exponential using exponent rules. Next, factor out ex. Since ex does not contain h, we can move it outside the limit.

  3. 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. Table Graph x -.1 -.01 -.001 .001 .01 .1 y .95 .995 .999 1.0005 1.005 1.05 At x = 0, f(0) appears to be 1. As x approaches 0, y approaches 1.

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

  5. Example 1: Find the derivative of f(x) = x2ex . Solution: Do you remember the product rule? You will need it here. Product Rule: (1st)(derivative of 2nd) + (2nd)(derivative of 1st) Factor out the common factor xex.

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

  7. 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.

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

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

  10. 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).

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

  12. 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:

  13. Example 5: Differentiate the function Solution: Using the quotient rule Keep in mind that the derivative of e-t is e-t(-1) or -e-t Recall that e0 = 1.

  14. Find the derivative of . Click on the button for the correct answer.

  15. 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. Return

  16. Good work!! Here is the derivative in detail.

  17. Example 6: A quantity growing according to the law where Q0 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.

  18. Example 7: Find the inflection points of Solution: We must use the 2nd 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.

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

  20. 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 Exponential Function Rule 2: If f(x) is a differentiable function then

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