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Differential calculus is a powerful mathematical tool
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for analyzing how things change.
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The handles of this tool
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are a few simple rules for calculating derivatives.
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Somewhere around 600 BC,
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Someone discovered that if you want to make
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pleasant-sounding chords on a string instrument
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then the lengths of the strings ought to be a ratio of simple numbers, like 1:2, or 2:3, and so on.
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These are called the 'Pythagorean harmonics'.
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The discovery was very important because it was the very first discovery
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of a connection between mathematics and the physical world.
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Unfortunately, that connection was forgotten,
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and had to be rediscovered slowly and painfully
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thousands of years later.
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One person would certainly grasp it though, was Galileo Galilei.
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In that case, I want to read to you something that Galileo wrote.
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This book was published in Rome, in 1623,
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and is called "IL SAGGIATORE"
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which is usually translated to mean "The Assayer."
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But I preferred to call "The Experimentalist"
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because I think that's more like what Galileo had in mind.
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Galileo had this nasty habit
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of making his famous remarks in Italian.
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But I'll translate it for you as I go along.
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He said: "True knowledge is written in this enormous book
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which is continously open before our eyes.
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I speak of the universe.
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But one can't understand it unless first
one learns to understand the language
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and recognized the characters in which it is written.
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It's written in a language of mathematics."
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So, to prepare to read the book of the universe,
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we must first learn its symbols and the vocabulary,
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of the language of mathematics.
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It's a language of precision
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of poetry and even of music.
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For a long time, physicists argued the language of mathematics.
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And since somewhere around 600 BC,
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so have musicians.
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As with just about all languages, including music.
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Mathematics has its own vocabulary,
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its own rules and samples,
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its precision and elegance,
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its poetry and its history.
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One part of that history was Galileo Galilei,
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who was something of a nonconformist.
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It was a trait he inherited from his father Vincenzo,
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who was an accomplished musician.
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Musically, Vincenzo refused to be bound by traditional forms.
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A position that would become the family trademark.
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He wrote a book composing theses of the Pythagorean Harmonics
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among his musical contemporaries.
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He considered the Ancient Greek chords too simple
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for the complex musical structures of the Italian Renaissance.
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Later, like father like son,
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Galileo considered the Greek standards of
mathematics too simple to express his ideas.
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He created the science of kinematics,
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a branch of mechanics dealing with motion and the abstract.
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And if any abstract idea is to be expressed properly,
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it needs appropriate language.
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The concepts and symbols that give an idea its meaning and value
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As advanced as it was,
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Galileo's new science of motion
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was still rooting in the soil of the Ancient Greek candlelight.
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And for anything entirely new to
bloom in the fields of mathematics and science,
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scholars needed a language more sophisticated
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than the one that had been spoken since Archimedes and Euclid.
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In other words, after Galileo,
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physics needed an advanced language.
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Fewer than 25 years after his death,
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that very language would be discovered,
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and in one form or another, spoken ever after.
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It would come to be called "Differential Calculus."
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Differential calculus is very powerful.
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And as with any language,
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it derives its power from the idea behind it:
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the derivative.
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The derivative is to kinematics what the wheel is to travel.
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A simple, yet very effective means
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for getting from one place to another.
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And, to get just the right perspective of what else the derivative is
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Nothing works better than a little exercise.
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To begin with, a derivative doesn't apply
only to a body in horizontal motion,
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nor for that matter, only to a body in vertical motion.
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Up, down, any direction.
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A derivative is the rate of change of any function,
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at any exact point or estimate
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As illustrated by Galileo's law of falling bodies,
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Speed is the derivative of distance.
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But it's more than that.
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A derivative can represent the rate of change of anything.
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For example,
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The population density of dolphins
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in relation to increasing or decreasing water temperature.
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Or the rate of change in the volume of a balloon
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versus its surface area.
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Or even the rate of change in the
price of a pizza with respect to its size.
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Clearly then, the concept of the derivative goes far and wide.
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But the mechanical process of the derivative,
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differential calculus, needs a practical approach.
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Or the concept goes nowhere fast.
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Eventually, without the rules of differentiation,
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the simple concept of the derivative becomes an uphill struggle.
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In the long run, it helps to pick up
a few more definitions along the way.
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So before it's too late to turn back,
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consider the factor of steepness.
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On the incline,
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the steepness is the ratio of the change in elevation,
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the change in horizontal distance.
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This ratio, a number, is called 'the slope'.
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For example,
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Suppose the elevation of an incline increases 15m every 100m
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The rider moves upward 15, horizontally 100.
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The slope is 0.15
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A hill with a slope of 0.3
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is twice as steep as one with the slope of 0.15
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The bigger the slope, the steeper the hill.
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When the slope is large, it's no small feat to get to the top.
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When the slope is next to nothing, near 0, it's easy-going.
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And when the slope is negative, it's downhill all the way.
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Although mathematics can be easy-going,
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it does have its peaks and valleys
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And it always has.
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Nobody knows who first asked
the best way to get from here to there.
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But the answer, in algebraic terms,
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was first offered by a French mathematician named Fermat.
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In 1629,
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He came up with the idea of finding the
tangent line to an arbitrary point on a curve.
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In 1638,
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Fermat shared his discovery with his friend and rival, René Descartes.
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who had his own method for finding tangents to algebraic curves.
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Many of these mathematical ideas,
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especially those of Fermat
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were further developed by Wilhelm Leibniz and Isaac Newton
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into a general and systematic method of mathematical analysis:
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Differential Calculus.
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Setting history aside, at least for the present,
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Some timely questions remain.
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For example, on a smoothly changing curve,
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there's a constantly changing slope.
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How then, in today's language, is a slope calculated
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at any given point?
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To determine the slope at a particular point,
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Here for example, simple take another point on the hill
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it doesn't matter where.
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Now connect the two points with a straight line.
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That line is called a chord.
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And its slope depends on the location of the second point.
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If the first and second point are reasonably close,
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The chord is a reasonably good approximation of the bike's path.
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Now, move the second point closer to the first.
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Move it even closer.
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The slope is a number and as the points get closer together,
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The number gets closer to a certain value.
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It's reasonable to call that number 'the slope of the hill' at that point.
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The line with that slope through the point is called 'the tangent line.'
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And the tangent line is just what the chord turns into
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as the points get closer together.
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And the slope of the tangent line at that point
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is the slope of the hill.
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Instantaneous speed can be calculated along the same lines.
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Galileo's law of falling bodies apply here to a rather relative one.
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It's more than an experiment in detail.
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It's differential calculus to the rescue.
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Change in distance is divided by change in time.
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The ratio is the average speed during a given time interval.
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When that time shrinks to zero,
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the limiting value of the average speed is the instantaneous speed.
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Change in elevation is divided by change in horizontal distance.
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The result is the slope of the chord joining 2 points.
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When the horizontal distance shrinks to zero,
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the limiting value of the slope of the chord is the slope at that point.
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Differentiation could overcome calculations differ
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but not the essential concept nor the process.
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Speed is the derivative of distance with respect to time.
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Slope is the derivative of elevation with respect to horizontal distance.
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In any case, a derivative is what happens to a quotient.
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A ratio of 2 numbers as both top and bottom shrinks to zero.
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Before they actually reach zero,
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small numbers are marked by the Greek letter Δ (delta)
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Δy is the small change in y
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Δx, a small change in x
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So, Δy/Δx is merely a ratio of 2 small numbers.
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When the small numbers shrinks to zero,
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that ratio becomes a derivative.
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And the deltas become a new symbol
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dy/dx, the symbol of the derivative.
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Which means, the derivative with respect to x of the quantity y.
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Once this simple mechanics are mastered,
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finding the derivative for just about anything
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is no harder than flipping a switch.
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The derivative of a function is a slope of its tangent at each point.
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The derivative of a function is itself a function.
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If the function is linear, the slope is constant.
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And the derivative is just that constant.
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If y=sin(x), then dy/dx=cos(x).
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If y=cos(x), then dy/dx=-sin(x).
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Taking derivatives takes a little practice,
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but it's well worth the effort.
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And, considering any number of contemporary derivative machines,
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it has become a modern practice.
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A speedometer is a derivative machine.
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It measures the derivative of the distance traveled,
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at each instant along the way.
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The rate of change of position is the instantaneous speed
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expressed at miles per hour (MPH).
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Of course when the vehicle isn't moving,
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no distance is being traveled.
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Here, the position is constant.
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And the derivative of a constant is zero.
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Mathematics is a language built on grammatical structure
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A collection of rules that both builds and breaks down
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the composition of the task at hand,
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whenever the masterwork.
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From building a house to composing a symphony.
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The most complicated task can be broken down in much the same way.
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Newton and Leibniz developed the tools of calculus
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that permit the most complex functions to be differentiated
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by breaking it down into simple parts.
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One of the basic rules of differentiation is the sum rule.
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Suppose one painter can paint 90m² of wall per hour
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And the other 100.
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Those are the rates at which areas of the wall are changing color
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In other words, they are derivatives.
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Therefore, every hour, 190m² of wall are being painted altogether.
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That's how the sum rule works.
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The derivative of the sum
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is the sum of the derivatives.
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Another handy tool is the product rule.
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which is used to find the derivative of the product of two functions.
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For example, the area of any board
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is the product of its length times its width.
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If the length is shortened,
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the change in area is the width times the change in length.
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If the width is reduced,
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the change in area is the length times the change in width.
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The total change in area is the sum of these.
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That's true of the carpenter's product
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and is just as true in the language of differential calculus.
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The derivative of the product yz
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is y times the derivative of z
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plus z times the derivative of y.
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Using this rule, it's possible to find the derivative of x²
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Or of x³
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Or of any power of x
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The derivative of xⁿ is n.x^(n-1)
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Often when an operation is depended on another.
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For example, suppose the vehicle has a fuel efficiency of 17 miles per gallon.
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That, too, is a derivative.
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If y is the distance traveled, and x the amount of fuel consumed
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Then, 17 miles per gallon equals dy/dx.
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Say it uses 2 gallons every hour,
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2 gallons per hour equals dx/dt.
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A vehicle's speed in miles per hour is equal to
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the miles per gallon it gets times the gallons per hour it uses.
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This is the chain rule.
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It's used when y depends on x
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and x depends on t.
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The sum rule
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The product rule
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And the chain rule
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These 3 rules represent the grammar of differential calculus.
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And the value of differential calculus
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can be seen in the variety of its applications.
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For example, when a rocket moves with displacement s and time t
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The derivative of the displacement is the velocity
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positive for upward motion
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and negative for downward motion.
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The derivative of the velocity is the acceleration
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which is the same as taking the derivative of a derivative.
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That is, the second derivative of s.
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The acceleration is caused by the firing of the rocket.
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The rules of differential calculus
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and their applications to physics
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This stands as the solo instrument
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and play in the art and science of mathematics.
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Working together, harmonizing, they can blend invidual notes or numbers
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into an exquisite harmony.
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I just got a letter from a musician named Albert Einstein.
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He sent it in 1912.
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The mail was lost some times.
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But that's not the reason that I just got it,
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the reason is because he didn't send it to me, he send it to a friend of his
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named Arnold Sommerfeld. And I just got it in the library.
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Anyway, I wanted to read to you a little bit of what he wrote.
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He said: "I occupy myself exclusively with the problem of gravitation.
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And now, believe I'll overcome all difficulties.
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Because one thing is certain
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I've become imbued with a great respect for mathematics
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The subtle parts of which, in my ignorance,
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I had, until now, regarded as pure luxury."
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Einstein worked on gravitation for 4 more years
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And what came out is called "the general theory of relativity"
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It is the most fiercely difficult mathematical theory in all of physics.
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So what did Einstein mean by saying that
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the subtle parts of mathematics had seemed a luxury?
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Did he really believe that he was going to get along without doing any calculations?
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Well of course not.
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The point is that, physicists have a certain arrogance about mathematics.
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For example, in today's lecture,
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You must have gotten the impression that all you have to do
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is follow some simple rules
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and you can take the derivative of any function in the world.
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Well that's not quite true.
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Suppose you had a function
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which looks like an Egyptian pyramid.
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Well, it's easy to see what the slope is here.
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And it's easy to see what the slope is here.
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But right here at the peak, you'd be in trouble.
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Because it has no slope at that point.
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The function has no derivative at that point.
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Now I never told you anything that would lead you to believe
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that that could ever happen.
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You see, for physicists, mathematics is just the tool
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It's to be used in order to accomplish something else.
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But a real mathematician
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is the guardian of precision and clarity of thought
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What interests the mathematician is the mathematics itself.
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When a mathematician makes a statement about derivatives,
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the statement takes into account every exception
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no matter how bizarre or unusual, like the peak of the pyramid.
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That's the kind of subtlety
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that Einstein was worried about.
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I'll see you next time.
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Subtitle created by Tran Nguyen Phuong Thanh - 2013.