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Proof: d/dx(sqrt(x))

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    So I've been requested to do
    the proof of the derivative of
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    the square root of x, so I
    thought I would do a quick
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    video on the proof of the
    derivative of the
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    square root of x.
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    So we know from the definition
    of a derivative that the
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    derivative of the function
    square root of x, that is equal
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    to-- let me switch colors, just
    for a variety-- that's equal to
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    the limit as delta
    x approaches 0.
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    And you know, some people
    say h approaches 0,
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    or d approaches 0.
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    I just use delta x.
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    So the change in x over 0.
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    And then we say f of x
    plus delta x, so in this
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    case this is f of x.
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    So it's the square root of x
    plus delta x minus f of x,
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    which in this case it's
    square root of x.
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    All of that over the change
    in x, over delta x.
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    Right now when I look at that,
    there's not much simplification
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    I can do to make this come out
    with something meaningful.
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    I'm going to multiply the
    numerator and the denominator
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    by the conjugate of the
    numerator is what
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    I mean by that.
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    Let me rewrite it.
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    Limit is delta x approaching
    0-- I'm just rewriting
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    what I have here.
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    So I said the square root
    of x plus delta x minus
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    square root of x.
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    All of that over delta x.
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    And I'm going to multiply
    that-- after switching colors--
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    times square root of x plus
    delta x plus the square root of
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    x, over the square root of x
    plus delta x plus the
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    square root of x.
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    This is just 1, so I could of
    course multiply that times-- if
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    we assume that x and delta x
    aren't both 0, this is a
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    defined number and
    this will be 1.
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    And we can do that.
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    This is 1/1, we're just
    multiplying it times this
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    equation, and we get limit
    as delta x approaches 0.
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    This is a minus b
    times a plus b.
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    Let me do little aside here.
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    Let me say a plus b times
    a minus b is equal to a
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    squared minus b squared.
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    So this is a plus b
    times a minus b.
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    So it's going to be
    equal to a squared.
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    So what's this quantity squared
    or this quantity squared,
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    either one, these are my a's.
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    Well it's just going
    to be x plus delta x.
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    So we get x plus delta x.
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    And then what's b squared?
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    So minus square root of
    x is b in this analogy.
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    So square root of x
    squared is just x.
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    And all of that over delta
    x times square root of x
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    plus delta x plus the
    square root of x.
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    Let's see what
    simplification we can do.
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    Well we have an x and
    then a minus x, so those
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    cancel out. x minus x.
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    And then we're left in the
    numerator and the denominator,
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    all we have is a delta x here
    and a delta x here, so let's
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    divide the numerator and the
    denominator by delta x.
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    So this goes to 1,
    this goes to 1.
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    And so this equals the limit--
    I'll write smaller, because I'm
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    running out of space-- limit as
    delta x approaches 0 of 1 over.
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    And of course we can only do
    this assuming that delta--
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    well, we're dividing by delta
    x to begin with, so we know
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    it's not 0, it's just
    approaching zero.
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    So we get square root
    of x plus delta x plus
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    the square root of x.
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    And now we can just
    directly take the limit
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    as it approaches 0.
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    We can just set delta
    x as equal to 0.
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    That's what it's approaching.
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    So then that equals one
    over the square root of x.
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    Right, delta x is 0, so
    we can ignore that.
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    We could take the limit
    all the way to 0.
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    And then this is of course just
    a square root of x here plus
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    the square root of x,
    and that equals 1 over
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    2 square root of x.
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    And that equals 1/2x
    to the negative 1/2.
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    So we just proved that x to the
    1/2 power, the derivative of it
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    is 1/2x to the negative 1/2,
    and so it is consistent with
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    the general property that the
    derivative of-- oh I don't
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    know-- the derivative of x to
    the n is equal to nx to the n
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    minus 1, even in this case
    where the n was 1/2.
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    Well hopefully
    that's satisfying.
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    I didn't prove it for all
    fractions but this is a start.
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    This is a common one you
    see, square root of x, and
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    it's hopefully not too
    complicated for proof.
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    I will see you in
    future videos.
Title:
Proof: d/dx(sqrt(x))
Description:

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Video Language:
English
Team:
Khan Academy
Duration:
05:08

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