2-3: Product / Quotient Rules & Other Derivatives ©2002 Roy L. Gover Objectives: Learn and use the product & quotient rules. Derive derivatives of trignometric functions. Use higher-order derivatives.

Important Idea The derivative of the product is not the product of the derivatives.

The Product Rule The derivative of the product of two functions is the first function times the derivative of the second plus the second function times the derivative of the first.

The Product Rule Memorize

Example Find the derivative, if it exists: The product of two functions

Important Idea Be sure you simplify your answers by at least: combining like terms eliminating negative exponents

Try This Find the derivative: Can you use a method other than the product rule?

Example Find the derivative: Note the new notation for derivative In this example, you must use the product rule.

Try This Find the derivative:

Important Idea The derivative of a quotient is not the quotient of the derivatives.

The Quotient Rule The derivative of a quotient is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator all divided by the denominator squared.

The Quotient Rule lo D hi minus hi D lo over lo 2

Example Find the derivative using the quotient rule: Is there an easier way…

Important Idea Sometimes it is easier to re-write the function and find the derivative using rules other than the quotient rule.

Example Must use the quotient rule on this one…

Try This Find the derivative using the quotient rule and simplify your answer:

Try This Find an equation of the line tangent to s(t) at t =2:

Try This Find the derivative (hint: re- write and use the quotient rule):

Example Find:

Try This Find: Hint: write and use the quotient rule.

Do This Memorize the derivatives of the 6 trig functions on pages 120 & 131.

Try This Differentiate: What rule was used?

Definition If you take the derivative of a derivative, you get a higher-order derivative. The notation is: Second derivative:

Definition If you take the derivative of a derivative, you get a higher-order derivative. The notation is: Third derivative:

Important Idea The first derivative represents a rate of change. The second derivative represents the rate of change of the rate of change. In physics, the first derivative is velocity; the second derivative is acceleration.

Example The height above the ground of an object dropped from altitude is: Find the velocity and acceleration of the object after 10 seconds.

Lesson Close In words and without using your notes, what is the product rule? In words and without using your notes, what is the quotient rule?

Assignment 134/1-43 odd,51,53,86