# Product & Quotient Rules Higher Order Derivatives

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Product & Quotient Rules Higher Order Derivatives
Lesson 2.3

After this lesson, you should be able to:
Find the derivative of a function using the Product Rule Find the derivative of a function using the Quotient Rule Find the derivative of a trig function Find a higher-order derivative of a function

How would you put this rule into words?
Basic Rules Product Rule How would you put this rule into words?

The derivative of a product of 2 functions is equal to the first times the derivative of the second plus the second function times the derivative of the first.

Theorem 2.7 The Product Rule

Try Some Use additional rules to determine the derivatives of the following function

Try Some More Use additional rules to determine the derivatives of the following function a)

Try Some More Use additional rules to determine the derivatives of the following function b)

Try Some More Use additional rules to determine the derivatives of the following function b)

Try Some More Use additional rules to determine the derivatives of the following function b) NOTE!!! YOU MUST FACTOR COMPLETELY!!!!!!!!!! GET IT DOWN TO THE POINT WHERE YOU COULD SOLVE FOR THE ZEROS OF THE EQUATION!!!!!

How would you put this rule into words?
Basic Rule Quotient Rule How would you put this rule into words?

The derivative of a quotient of 2 functions is equal to the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator all over the square of the denominator.

Theorem 2.8 The Quotient Rule

Just Checking . . . Find the derivatives of the given functions

Just Checking . . . Find the derivatives of the given functions

Just Checking . . . Find the derivatives of the given functions

Just Checking . . . Find the derivatives of the given functions

Other Trig Derivatives
Now try it out

Theorem 2.9 Derivatives of Trigonometric Function

Other Trig Derivatives

JUST AN EXAMPLE OF HOW COMPLICATED THEY CAN GET
JUST AN EXAMPLE OF HOW COMPLICATED THEY CAN GET!!!! CAN WE SIMPLIFY FURTHER???

Derivatives of the remaining trig functions can be determined the same way.
p

Find Those High Orders Find the requested derivatives

Find each derivative.

Find the derivative.

Find the derivative.

Equation of a Horizontal Tangent Line
Find the equations of the horizontal tangent lines for

Equation of a Horizontal Tangent Line
Find the equations of the horizontal tangent lines for

Equation of the Tangent Line
Example: Write the equation of the tangent line at

Equation of the Tangent Line
Example: Write the equation of the tangent line at

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