The Quotient and Product RulesA story of f(x) and g(x)

A Fleet Street Production

Once upon a time long ago in the Ancient Once upon a time long ago in the Ancient World of Mathematics, two rival families of World of Mathematics, two rival families of functions ruled the advancing Kingdom called functions ruled the advancing Kingdom called Calculus. The first family of functions was the Calculus. The first family of functions was the Family of F. They had long been the supreme rulers Family of F. They had long been the supreme rulers of Calculus due to their great presence not only in of Calculus due to their great presence not only in Calculus, but also in other kingdoms like Algebra, Calculus, but also in other kingdoms like Algebra, Trigonometry, and Pre-calculus. In the recent years, Trigonometry, and Pre-calculus. In the recent years, however, as the Kingdom of Calculus was further however, as the Kingdom of Calculus was further advancing another family was expanding in its advancing another family was expanding in its existence, called the Family of G.existence, called the Family of G.

On one fateful summer day in the Kingdom, On one fateful summer day in the Kingdom, there was great news in the Family of F. Unto them there was great news in the Family of F. Unto them had been born two baby functions, two twin boys had been born two baby functions, two twin boys who were both quickly named f(x). The family’s joy who were both quickly named f(x). The family’s joy was soon dimmed, however, when the news of two was soon dimmed, however, when the news of two twin girls named g(x) was heard throughout the twin girls named g(x) was heard throughout the kingdom. Both families, adamant that the pride and kingdom. Both families, adamant that the pride and joy of their lives never be tainted by that of their joy of their lives never be tainted by that of their rival, vowed to raise their children completely apart rival, vowed to raise their children completely apart from each other.from each other.

Unfortunately, this isolation could not be Unfortunately, this isolation could not be preserved as both f(x) and f(x) and g(x) and g(x) preserved as both f(x) and f(x) and g(x) and g(x) became of age and attempted to wriggle away became of age and attempted to wriggle away from the iron grips of their parents. One day as from the iron grips of their parents. One day as both the f(x) twins were out on a journey to both the f(x) twins were out on a journey to discover their derivatives, they beheld the g(x) discover their derivatives, they beheld the g(x) twins who were on the same journey. Both sets of twins who were on the same journey. Both sets of twins immediately fell desperately in love and twins immediately fell desperately in love and vowed to get married regardless of the wishes of vowed to get married regardless of the wishes of their parents.their parents.

One couple of f(x) and g(x) were married in One couple of f(x) and g(x) were married in the church of multiplication, then becoming the church of multiplication, then becoming f(x)g(x). The second couple, having slightly more f(x)g(x). The second couple, having slightly more liberal religious beliefs, were married in the church liberal religious beliefs, were married in the church of division, thus becoming f(x)/g(x). Needless to of division, thus becoming f(x)/g(x). Needless to say, when their parents found out about the matter say, when their parents found out about the matter they were irate. The two couples were banished to they were irate. The two couples were banished to a subterranean prison from which they could never a subterranean prison from which they could never escape unless they found their derivatives.escape unless they found their derivatives.

Cold and tired in the underground forest, the Cold and tired in the underground forest, the couples tried and failed to find their derivatives. couples tried and failed to find their derivatives. They knew how individually, but they were They knew how individually, but they were completely lost and thought themselves doomed. completely lost and thought themselves doomed. However, one night from the corner of the prison However, one night from the corner of the prison they heard a low hum, the meditation of a very they heard a low hum, the meditation of a very wise function. They approached the old function wise function. They approached the old function with hesitance, but in desperation they asked him with hesitance, but in desperation they asked him for his great wisdom.for his great wisdom.

““My name is Natural Log,” he said with his My name is Natural Log,” he said with his eyes still closed, “and I have seen your troubles eyes still closed, “and I have seen your troubles and can help you.” Using the great force of his and can help you.” Using the great force of his mind, he inscribed in the dirt of the prison first the mind, he inscribed in the dirt of the prison first the Logarithmic proof of the product rule…Logarithmic proof of the product rule…

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……and the quotient rule.and the quotient rule.

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The couples were absolutely amazed, yet The couples were absolutely amazed, yet needing to apply their knowledge to be able to needing to apply their knowledge to be able to escape from the pit, they needed to know how to escape from the pit, they needed to know how to use it; so the wise and generous Natural Log gave use it; so the wise and generous Natural Log gave them a few examples.them a few examples.

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Product Rule

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Quotient Rule

So grateful for his generosity, the couples So grateful for his generosity, the couples asked the Natural Log what they could do for him asked the Natural Log what they could do for him in return. To f(x)/g(x), he asked that they go deep in return. To f(x)/g(x), he asked that they go deep into the forest to find tan x, sec x, cot x, and csc x into the forest to find tan x, sec x, cot x, and csc x to show them how to use the quotient rule to to show them how to use the quotient rule to discover their derivatives, which they did with discover their derivatives, which they did with alacrity. alacrity.

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Derivatives of Trig Functions

After they did so, f(x)/g(x) returned to their After they did so, f(x)/g(x) returned to their brother and sister who were busy holding back all brother and sister who were busy holding back all of the other imprisoned functions from mobbing of the other imprisoned functions from mobbing the Natural Log. Upon their arrival, a huge crowd the Natural Log. Upon their arrival, a huge crowd of functions with whole numbers in their of functions with whole numbers in their denominators surrounded f(x)/g(x) so they could denominators surrounded f(x)/g(x) so they could learn their derivatives and be free.learn their derivatives and be free.

In the midst of the commotion, the Natural In the midst of the commotion, the Natural Log’s booming voice brought the whole forest to Log’s booming voice brought the whole forest to silence. “You do not need the quotient rule to be silence. “You do not need the quotient rule to be derived,” he said in his wisdom. “If you can factor derived,” he said in his wisdom. “If you can factor out a constant from yourself, you may move it to out a constant from yourself, you may move it to the outside of the derivative, simply find your the outside of the derivative, simply find your derivative by normal practice, and multiply your derivative by normal practice, and multiply your derivative by that constant.” He used his derivative by that constant.” He used his perspicacious mind to give them an example of perspicacious mind to give them an example of this constant multiple rule in the soil.this constant multiple rule in the soil.

The Product Multiple RuleThe Product Multiple Rule

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““Great Natural Logarithm,” cried another one Great Natural Logarithm,” cried another one of the prisoners bound by an exponent, “give us a of the prisoners bound by an exponent, “give us a proof and a power rule!”proof and a power rule!”

““No proof,” responded the Natural Log. When No proof,” responded the Natural Log. When the hopeless cries began among the exponentially the hopeless cries began among the exponentially bound function, he said, “A Logarithmic proof is far bound function, he said, “A Logarithmic proof is far too complicated. You must simply have faith in this too complicated. You must simply have faith in this derivative and you shall be set free.” Discontented, derivative and you shall be set free.” Discontented, skeptical functions returned to the depths of the skeptical functions returned to the depths of the forest, but the hopeful remained. He wrote the forest, but the hopeful remained. He wrote the power rule in the ground followed by examples.power rule in the ground followed by examples.

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All of the functions were ecstatic and began All of the functions were ecstatic and began to run towards the exit. When the Natural Log did to run towards the exit. When the Natural Log did not follow, f(x)g(x) asked him, “Why do you not not follow, f(x)g(x) asked him, “Why do you not come with us? We can be free!”come with us? We can be free!”

With a knowing smile and his eyes still With a knowing smile and his eyes still closed he said, “My time has not yet come. Go and closed he said, “My time has not yet come. Go and be useful and I shall see you again in chapter five.” be useful and I shall see you again in chapter five.” With those final words, the Natural Log With those final words, the Natural Log disappeared.disappeared.

Proudly stating their derivatives upon their Proudly stating their derivatives upon their exit, all of the functions were set free from the exit, all of the functions were set free from the prison. However, leaning up against a tree in the prison. However, leaning up against a tree in the kingdom, a pretty little function was crying whose kingdom, a pretty little function was crying whose name was s(t).name was s(t).

“Why are you crying when you are free?” the “Why are you crying when you are free?” the liberated functions asked.liberated functions asked.

“Well, I’ve been confined to the World of “Well, I’ve been confined to the World of Calculus and am scheduled to leave for the Calculus and am scheduled to leave for the wonderful World of Physics. The only problem is,” wonderful World of Physics. The only problem is,” she whimpered, “I am a position function and I she whimpered, “I am a position function and I need to find my acceleration first!”need to find my acceleration first!”

““We can take your derivative!” excitedly We can take your derivative!” excitedly screamed the other functions, scribbling away on screamed the other functions, scribbling away on the ground to find her derivative. the ground to find her derivative.

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““Thank you,” she said politely before she Thank you,” she said politely before she began to cry again, “but a first derivative is only began to cry again, “but a first derivative is only velocity, and I have no idea how to find velocity, and I have no idea how to find accelerationacceleration!”!”

““We can take the derivative of velocity We can take the derivative of velocity then!” the crowds roared. So they derived the first then!” the crowds roared. So they derived the first derivative to get a second derivative, which derivative to get a second derivative, which represented acceleration.represented acceleration.

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““That is perfect!” she told them all. “Thank That is perfect!” she told them all. “Thank you so much for your abounding kindness. I shall you so much for your abounding kindness. I shall repay you somehow!” She instantly reached her repay you somehow!” She instantly reached her necessary acceleration and disappeared into the necessary acceleration and disappeared into the sky, illuminating it with great light until the arrival sky, illuminating it with great light until the arrival of the section of implicit differentiation. So f(x)g(x) of the section of implicit differentiation. So f(x)g(x) and f(x)/g(x) lived happily ever after and dedicated and f(x)/g(x) lived happily ever after and dedicated their lives to helping AP Calculus high school their lives to helping AP Calculus high school students in memory of the kindness of the Natural students in memory of the kindness of the Natural Log.Log.

By:

Elexis Ellis

Komal Patel

& Rebecca Jones