1. Calculus I (MAT 145) Dr. Day Monday February 18, 2019
Derivative Shortcuts (Ch 3)
Sums, Differences, Product, Quotients
Constants, Powers, Exponential Functions
Trig Functions
The Chain Rule!
Return Quiz #5
Monday, February 18, 2019
MAT 145

2. Warm up! . Calculate each derivative. Use correct notation!
Monday, February 18, 2019
MAT 145

3. Sums, differences, exponentials, & products of constants and functions
Monday, February 18, 2019
MAT 145

4. Derivatives of Trig Functions
Monday, February 18, 2019
MAT 145

5. Monday, February 18, 2019
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6. Monday, February 18, 2019
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7. Practice Derivative Rules
Monday, February 18, 2019
MAT 145

8. Using Derivative Patterns
For f(x) = 2x2 – 3x + 1:
Calculate f’(x).
Determine an equation for the line tangent to the graph of f when x = −1.
Determine all values of x that lead to a horizontal tangent line.
Determine all ordered pairs of f for which f’(x) = 1.
Monday, February 18, 2019
MAT 145

9. Using Derivative Patterns
MAT 145
Suppose s(x), shown below, represents an object’s position as it moves back and forth on a number line, with s measured in centimeters and x in seconds, for x > 0.
Calculate the object’s velocity and acceleration functions.
Is the object moving left or right at time x = 1? Justify.
Determine the object’s velocity and acceleration at time x = 2. Based on those results, describe everything you can about the object’s movement at that instant.
Write an equation for the tangent line to the graph of s at time x = 1.
Monday, February 18, 2019

10. Using Derivative Patterns
Determine the equation for the line tangent to the graph of g at x = 4.
Determine the equation for the line normal to the graph of g at x = 1.
At what points on the graph of g, if any, will a tangent line to the curve be parallel to the line 3x – y = –5?
Monday, February 18, 2019
MAT 145

11. Practice Derivative Rules - Answers
Monday, February 18, 2019
MAT 145

12. Practice Derivative Rules
Monday, February 18, 2019
MAT 145

13. Practice Derivative Rules - Answers
Monday, February 18, 2019
MAT 145

14. Derivatives of Composite Functions
Monday, February 18, 2019
MAT 145

15. Derivatives of Composite Functions (3.4)
Monday, February 18, 2019
MAT 145

16. Here's a function in a function And your job here is to find
THE CHAIN RULE
WORDS BY: JOHN A. CARTER
TUNE: "CLEMENTINE"
Here's a function in a function
And your job here is to find
The derivative of the whole thing
With respect to x inside.
Call the outside f of u
And call the inside u of x.
Differentiate to find df/du
And multiply by du/dx.
Use the chain rule.
Use the chain rule when e'er you find
The derivative of a function compositionally defined.
Monday, February 18, 2019
MAT 145

17. Derivatives of Composite Functions (3.4)
Monday, February 18, 2019
MAT 145

18. Derivatives of Composite Functions
Monday, February 18, 2019
MAT 145

19. Composite Functions
Monday, February 18, 2019
MAT 145