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Derivative as a concept | Derivatives introduction | AP Calculus AB | Khan Academy - YouTube
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you are likely already familiar with the
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idea of a slope of a line if you're not
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i encourage you to review it on khan
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academy but all it is it's describing
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the rate of change of a vertical
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variable with respect to a horizontal
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variable so for example here i have our
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classic y-axis in the vertical direction
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and x-axis in the horizontal direction
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and if i wanted to figure out the slope
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of this line i could pick two points say
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that point and that point
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i could say okay from this point to this
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point what is my change in x well my
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change in x would be this distance right
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over here
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change in x
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the greek letter delta this triangle
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here it's just shorthand for change so
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change in x
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and i could also calculate the change in
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y
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so
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this point going up to that point our
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change in y
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would be
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this right over here our change in y
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and then we would define slope or we
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have defined slope as change in y over
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change in x
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so slope
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is equal to the rate of change of our
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vertical variable over the rate of
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change of our horizontal variable it's
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sometimes described as rise over run
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and for any line it's associated with a
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slope because it has a constant rate of
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change if you took any two points on
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this line no matter how far apart or no
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matter how close together anywhere they
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sit on the line if you were to do this
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calculation you would get the same slope
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that's what makes it a line but what's
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fascinating about calculus is we're
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going to build the tools so that we can
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think about the rate of change not just
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of a line which we've called slope in
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the past we can think about the rate of
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change the instantaneous rate of change
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of a curve of something whose rate of
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change is possibly constantly changing
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so for example
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here's a curve where the rate of change
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of y with respect to x is constantly
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changing even if we wanted to use our
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traditional tools if we said okay we can
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calculate the average rate of change
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let's say between this point and this
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point well what would it be well the
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average rate of change between this
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point and this point would be the slope
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of the line that connects them
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so it'd be the slope of this line of the
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secant line but if we pick two different
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points we pick this point and this point
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the average rate of change between those
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points all of a sudden looks quite
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different it looks like it has a higher
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slope so even when we take the slopes
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between two points on the line the
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secant lines you can see that those
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slopes are changing
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but what if we wanted to ask ourselves
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an even more interesting question
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what is the
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instantaneous rate of change at a point
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so for example
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how fast is y changing with respect to x
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exactly at that point exactly when x is
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equal to that value let's call it x1
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well one way you could think about it is
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what if we could draw a tangent line to
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this point a line that just touches the
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graph right over there and we can
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calculate the slope of that line well
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that should be the rate of change at
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that point the instantaneous rate of
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change so in this case the tangent line
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might look something
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like that if we know the slope of this
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well then we could say that that's the
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instantaneous rate of change at that
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point why do i say instantaneous rate of
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change well think about the video on the
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sprinters the usain bolt example if we
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wanted to figure out the speed of usain
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bolt at a given instant well maybe this
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describes his position with respect to
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time if y was position and x is time
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usually you would see t is time but
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let's say x is time so then if we're
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talking about right at this time we're
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talking about the
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instantaneous rate
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and this idea is the central idea of
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differential calculus and it's known as
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a derivative
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the slope of the tangent line
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which you could also view as the
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instantaneous
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rate of change i'm putting exclamation
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mark because it's so conceptually
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important here
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so how can we denote a derivative
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one way is known as leibniz's notation
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and leibniz is one of the fathers of
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calculus along with isaac newton and his
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notation you would denote the slope of
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the tangent line as
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equaling d y
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over dx
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now why do i like this notation because
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it really comes from this idea of a
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slope which is change in y over change
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in x
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as you'll see in future with videos one
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way to think about the slope of the
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tangent line is well let's calculate the
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slope of secant lines let's say between
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that point and that point but then let's
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get even closer say that point in that
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point and then let's get even closer in
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that point in that point and then let's
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get even closer and let's see what
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happens as the change in x approaches
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zero
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and so using these d's instead of deltas
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this was leibniz's way of saying hey
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what happens if my changes in say x
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become close to zero so this idea this
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is known as sometimes differential
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notation leibniz's notation is instead
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of just changing y over change in x
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super small changes in y for a
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super small change in x especially as
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the change in x approaches zero and as
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you will see that is how we will
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calculate the derivative
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now there's other notations
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if this curve is described as y is equal
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to
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f of x
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the slope of the tangent line at that
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point could be denoted as equaling
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f
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prime
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of x1
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so this notation it takes a little bit
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of time getting used to the lagrange
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notation
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it's saying f prime is representing the
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derivative it's telling us the slope of
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the tangent line for a given point
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so if you input an x into this function
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into f you're getting the corresponding
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y value if you input an x into f prime
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you're getting the slope of the tangent
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line at that point
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now another notation that you'll see
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less likely in a calculus class but you
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might see in a physics class is the
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notation
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y with a dot over it so you could write
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this as y
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with a dot over it which also denotes
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the derivative you might also see y
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prime this would be more common in a
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math class
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now as we march forward in our calculus
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adventure we will build the tools to
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actually calculate these things and if
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you're already familiar with limits they
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will be very useful as you can imagine
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because we're really going to be taking
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the limit of our change in y over change
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in x as our change in x approaches zero
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and we're not just going to be able to
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figure it out for a point we're going to
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be able to figure out general equations
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that describe the derivative for any
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given point so be very very excited
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