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2.3 Product and Quotient Rules and Higher-Order Derivatives
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Theorem 2.7 The Product Rule
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Using the Product Rule Find the derivative of:
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Using the Product Rule Find the derivative of:
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Using the Product Rule Find the derivative of:
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Theorem 2.8 The Quotient Rule
Copyright © Houghton Mifflin Company. All rights reserved.
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Using the Quotient Rule Find the derivative of
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Using the Quotient Rule Find the derivative of
Begin by rewriting the function:
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Using the Constant Multiple Rule
Original Function: Rewrite: Now differentiate: And simplify:
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Using the Constant Multiple Rule
Original Function: Rewrite: Now differentiate: And simplify:
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Using the Constant Multiple Rule
Original Function: Rewrite: Now differentiate: And simplify:
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Using the Constant Multiple Rule
Original Function: Rewrite: Now differentiate: And simplify:
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Theorem 2.9 Derivatives of Trigonometric Function
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Applying the Quotient Rule:
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Differentiating Trigonometric Functions
Derivative:
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Different Forms of a Derivative
First form: Second Form: To show that these two derivatives are equal
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To show that these two derivatives are equal:
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Higher-Order Derivatives
Position Function: Velocity Function: Acceleration Function:
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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.
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Finding Acceleration Due to Gravity
To find the acceleration, differentiate the position function twice. Position function: Velocity function: Acceleration function:
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