1. 2.3 Product and Quotient Rules and Higher-Order Derivatives

2. Theorem 2.7 The Product Rule
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3. Using the Product Rule Find the derivative of:

4. Using the Product Rule Find the derivative of:

5. Using the Product Rule Find the derivative of:

6. Theorem 2.8 The Quotient Rule
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7. Using the Quotient Rule Find the derivative of

8. Using the Quotient Rule Find the derivative of
Begin by rewriting the function:

9. Using the Constant Multiple Rule
Original Function:
Rewrite:
Now differentiate:
And simplify:

10. Using the Constant Multiple Rule
Original Function:
Rewrite:
Now differentiate:
And simplify:

11. Using the Constant Multiple Rule
Original Function:
Rewrite:
Now differentiate:
And simplify:

12. Using the Constant Multiple Rule
Original Function:
Rewrite:
Now differentiate:
And simplify:

13. Theorem 2.9 Derivatives of Trigonometric Function
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14. Applying the Quotient Rule:

15. Differentiating Trigonometric Functions
Derivative:

16. Different Forms of a Derivative
First form:
Second Form:
To show that these two derivatives are equal

17. To show that these two derivatives are equal:

18. Higher-Order Derivatives
Position Function:
Velocity Function:
Acceleration Function:

19. Finding the Acceleration Due to Gravity
Because the moon has no atmosphere, a falling object on the moon encounters no air resistance. In 1971, astronaut David Scott demonstrated that a feather and a hammer fall at the same rate on the moon. The position function for each of these falling objects is given by:
where s(t) is the height in meters and t is the time in seconds.

20. Finding Acceleration Due to Gravity
To find the acceleration, differentiate the position function twice.
Position function:
Velocity function:
Acceleration function: