Rules for Differentiation Finding the Derivative of a Product of Two Functions. What does this equation of f '(
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1 Rules for Differentiation Finding the Derivative of a Product of Two Functions Rewrite the function f( = ( )( + 1) as a cubic function. Then, find f '(. What does this equation of f '( represent, again? Two men, Isaac Newton and Gottfried Leibniz, are credited for developing the study of calculus. In 167, Leibniz published an article in which he derived what we know today as the Product Rule of Differentiation. Let s write this rule together in the bo below. Product Rule of Differentiation To show that this rule works, let s apply this rule to the function f( = ( )( + 1) that we rewrote and differentiated as a polynomial above. Students often wonder why this rule is so important if we could just rewrite as a polynomial and easily differentiate it. The answer to that question is simple. If it is possible to rewrite as a polynomial, always do so. But in the case of the function g( sin, there is no way to rewrite as a polynomial. Apply the product rule to find the slope of the normal line to the graph of g( sin when = π. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 10 Mark Sparks 01
2 Use the product rule to find the derivative of each of the following functions. f ( g ( f ( sin h( ( ) cos g ( sin h( sin cos Find the equation of the line tangent to the graph of g( t) t cos t when t =. 6 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 11 Mark Sparks 01
3 There is a very valuable lesson that we must learn when we are introduced to the product rule. On page 11, you were asked to find f '( by applying the product rule to the function f (. In the space below, write the result that you obtained. Given the function ( f. Rewrite the function in polynomial form. Then, find f '(. What is the lesson to be learned from the algebraic analysis above? If g ( ( )( 1)( ), what is the slope of the normal line to the graph of g( when =? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 1 Mark Sparks 01
4 Below are graphs of two functions f( and g(. Let P( f ( g( and let R( g(. Use the graphs to answer the questions that follow. Graph of f( Graph of g( If g '( 4), what is the value of P '( 4)? If R '( ) = 0, what is the value of g '( )? Find the equation of the line tangent to the graph of P( when = 4. Find the equation of the line tangent to the graph of R( when =. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 1 Mark Sparks 01
5 Let f( and g( be differentiable functions such that the following values are true. f( g( f '( g '( Estimate the value of f '(.5). If q( f ( 4g(, what is the value of q '(4)? If p( f ( g(, what is the value of p '()? Find the equation of the line tangent to the graph of v( f ( when = 1. If k( f ( g(, what is the value of k '()? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 14 Mark Sparks 01
6 Rules for Differentiation Finding the Derivative of a Quotient of Two Functions Rewrite the function f ( as a function in polynomial form. Then, find f '(. Just as Leibniz was the first to publish a proof of the Product Rule for Differentiation, Isaac Newton was the first to publish a proof of the Quotient Rule of Differentiation using the limit definition of the derivative. Let s write this rule together in the bo below. Quotient Rule of Differentiation To show that this rule works, let s apply this rule to the function and differentiated as a polynomial-form above. f ( that we rewrote 1 Find the equation of the tangent line drawn to the graph of g( when =. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 15 Mark Sparks 01
7 We will now use the quotient rule to derive the derivative formulas for the remaining trigonometric functions. Rewrite each function in terms of sine and/or cosine and differentiate using the Quotient Rule. f ( ) tan f ( ) cot f ( ) sec f ( ) csc Find the equation of the normal line drawn to the graph of f ( ) when. cos Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 16 Mark Sparks 01
8 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 17 Mark Sparks 01 Find the derivative of each of the functions below by applying the quotient rule. ) ( f tan ) ( g cos 1 sin ) ( h 5 ) ( 1 f
9 f ( 1 Similar to the Product Rule, there is a very valuable lesson that we must learn when we are introduced to the quotient rule. 6 Given the function f (. Find 4 f '( by applying the quotient rule. 6 Given the function f (. Simplify 4 f (. Then, find f '( by applying the quotient rule. What is the lesson to be learned from the algebraic analysis above? Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 18 Mark Sparks 01
10 Below are graphs of two functions f( and g(. Let to answer the questions that follow. Graph of f( f ( sin P( and let R(. Use the graphs g( f ( Graph of g( Find P '(5). Find the equation of the line tangent to the graph of P( when = 5. Find ) '(0 R Find the equation of the line tangent to the graph of R( when = 0. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 19 Mark Sparks 01
11 Let f( and g( be differentiable functions such that the following values are true. f( g( f '( g '( Estimate the value of g '(.5). g( If p(, what is the value of p '(4)? What f ( does this value say about the graph of p( when = 4? Give a reason for your answer. f ( If q (, what is the value of q '()? g( Find the equation of the line tangent to the graph of v( when =. g( Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 0 Mark Sparks 01
12 Rules for Differentiation Finding the Derivative of a Composite Function Rewrite the function f ( ( ) as a function in polynomial form. Then, find f '(. Leibniz was the first of the two great calculus developers to use the Chain Rule to differentiate composite functions. Let s write this rule together in the bo below. Chain Rule of Differentiation of Composite Functions To show that this rule works, let s apply this rule to the function differentiated as a polynomial-form above. f ( ( ) that we rewrote and Find the slope of the normal line to the graph of 5 f ( ) sin when. 6 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 1 Mark Sparks 01
13 Find the derivative of each of the following functions by applying the chain rule. f ( g ( 5 h ( ( ) F 5 ( G( cos h ( sin ( 1) Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page Mark Sparks 01
14 Now that you know THE BIG THREE rules of differentiation product, quotient, and chain let s see how the three can be incorporated with each other. Find the derivative of each of the following functions. f ( 5 g ( sin 1 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page Mark Sparks 01
15 5 h ( Given the graph of H( pictured to the right, find the equation of the tangent line to the graph of P( H( when = 4. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 4 Mark Sparks 01
16 Let f( and g( be differentiable functions such that the following values are true. f( g( f '( g '( Is the graph of h( f ( g( ) increasing, decreasing or at a relative maimum or minimum when =? Give a reason for your answer. If p( g(, what is the value of p '(1)? If q( f ( g(, what is the value of q '(4)? What does this value say about the graph of q( when = 4? Give a reason for your answer. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 5 Mark Sparks 01
17 Problems to Discuss before Quiz #4 Problem #1 Find the following limit. Eplain the reasoning that you used to arrive at your answer. cos( h) cos lim h 0 h Problem # Find the equation of the tangent line to the graph of the given function when f ( cos. Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 6 Mark Sparks 01
18 Problem # Find the equation of the normal line to the graph of the function below when =. f ( Problem #4 At what point on the graph of the function f ( is the normal line perpendicular to the line defined by the equation y 1? 4 Daily Lessons and Assessments for AP* Calculus AB, A Complete Course Page 7 Mark Sparks 01
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