1. B.1.6 – DERIVATIVES OF EXPONENTIAL FUNCTIONS
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B.1.6 – DERIVATIVES OF EXPONENTIAL FUNCTIONS
Calculus - Santowski
CALCULUS - SANTOWSKI

2. Factor ex – e2x Factor e2x – ex Factor Simplify (2x)(22)
FAST FIVE
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Factor ex – e2x
Factor e2x – ex
Factor
Simplify (2x)(22)
Solve xe3x – 3xex = 0
Evaluate the limit
Calculus - Santowski

3. LESSON OBJECTIVES
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(1) Investigate the derivative of exponential functions using a variety of methods
(2) Develop an algebraic derivation of the derivative of an exponential function
(3) Apply the various differentiation rules to differentiate exponential functions
Calculus - Santowski

4. (A) EXPLORATION – PART 1
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You are now pursuing derivatives of other functions (not just power functions)
To begin our study of derivatives of exponential functions, let’s explore a bit first
1. Sketch y = bx Each partner at the table will use a different value for b
2. PREDICT the features of the graph of its derivative by answering the following Q
(a) Identify the intervals of increase and decrease
(b) identify the critical values
(c) From this information (and knowing what each means about the derivative), prepare a hand drawn graph of the derivative
Calculus - Santowski

5. (B) EXPLORATION – PART 2
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We will go back to our "first principles" - that being the idea that we can determine instantaneous rates of changes using tangent lines
(1) Use GDC to draw the tangent lines at various x values
(2) Record the slopes of the tangent lines on a table.
(3) Prepare a scatter plot from the table of values.
(4) Describe scatter plot
Calculus - Santowski

6. EXPLORATION – PART 3 Now let’s use graphing technology:
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Now let’s use graphing technology:
Use the TI-89 to directly and immediately prepare a graph of the derivative of y = bx.
What is the derivative of y = bx?
Confirm that your equation for the derivative is correct (and show/explain how you confirmed this.)
Calculus - Santowski

7. EXPLORATION – PART 4
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Now we will use algebra to PROVE that our observations were correct.
So we go back to our limit definition of a derivative:
Our definition is:
So work with it ……
Calculus - Santowski

8. DERIVATIVE OF EXPONENTIAL FUNCTIONS
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Calculus - Santowski

9. (B) INVESTIGATING THE LIMITS
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Investigate lim h0 (2h – 1)/h numerically with a table of values
x y
undefined
And we see the value of as an approximation of the limit
Investigate lim h0 (3h – 1)/h numerically with a table of values
x y
undefined
And we see the value of as an approximation of the limit
Calculus - Santowski

10. (B) INVESTIGATING THE LIMITS
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Investigate lim h0 (4h – 1)/h numerically with a table of values
x y
undefined
And we see the value of as an approximation of the limit
Investigate lim h0 (eh – 1)/h numerically with a table of values
x y
undefined
And we see the value of as an approximation of the limit
Calculus - Santowski

11. (C) SPECIAL LIMITS - SUMMARY
Is there a pattern to these numbers  the number (coming from base 2), (coming from base = 3), (base 4)
To explore, we can rewrite ax in base e as e(lna)x
So if d/dx ex was ex, then d/dx e(lna)x must be e(lna)x times lna (by the chain rule)
And so: ln(2) = 0.693
And so: ln(3) =
And so: ln(4) = 1.386
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Calculus - Santowski

12. (D) DERIVATIVES OF EXPONENTIAL FUNCTIONS - SUMMARY
The derivative of an exponential function was
Which we will now rewrite as
And we will see one special derivative  when the exponential base is e, then the derivative becomes:
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Calculus - Santowski

13. (E) EXAMPLES – DIFFERENTIATION OF EXPONENTIAL FUNCTIONS
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(1) Find the derivative of y = e3x
(2) Find the derivative of y = 3x2e2x
(3) Differentiate
(4) Differentiate
(5) Differentiate
(6) Differentiate
Calculus - Santowski

14. (F) EXAMPLES - APPLICATIONS
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1. Find the equation of the tangent line to the curve y = 1 + xe2x at x = 0
2. Find the intervals of increase/decrease for the function f(x) = x2e-x
Calculus - Santowski

15. (G) INTERNET LINKS
Calculus I (Math 2413) - Derivatives - Derivatives of Exponential and Logarithm Functions from Paul Dawkins
Visual Calculus - Derivative of Exponential Function
From pkving
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Calculus - Santowski

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Calculus - Santowski