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www.mathcentre.ac.uk/.../8.9a%20Use%20table%20derivs%2012%20functs.mp4

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    We're now going to extend our
    table of derivatives by looking
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    at functions such as tanks sex,
    arcsine X, or sign minus one X&A
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    to the power X, where a is a
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    positive constant. Let's start
    with the table with our
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    function. F of X.
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    And our derivative.
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    Either referred to as DF, DX,
    or F dashed of X.
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    And let's
    start with
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    tonics.
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    Sex.
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    Call Tex.
    And Cosec X.
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    And the first thing they want to
    want to do is to actually write
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    these in terms of sine and
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    cosine. So tonics we can write
    a sign X divided by Cos X.
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    Sex is 1 divided by Cos X.
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    Call Tex is Arcos X divided
    by sine X and cosec, X
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    is 1 divided by sine X.
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    So let's have a look now at
    finding the derivatives of some
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    of these. Let's
    start with tonics
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    Y equals TAN X.
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    Now we've already written tonics
    in terms of sine and cosine, so
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    let's write it a sign X divided
    by cosine X.
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    And this is as if we had
    U&V where you is a function of
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    X&V is a function of X.
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    So we have a quotient.
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    So we're going to use our rule
    for differentiating quotients.
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    Do Y by ZX equals V
    times? Do you buy the X
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    minus use times DV Pi DX?
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    All divided by V squared.
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    So in this case, IQ equals
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    sign X. So do you buy the
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    X? It was cool sex.
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    RV. It's called
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    sex. So 2 feet by TX.
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    Equals minus sign X.
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    Let's substitute now.
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    Into our derivative do I buy DX?
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    Is it cool to V which is Cos X?
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    Multiplied by do you buy DX,
    which is cause X?
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    Minus.
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    You which is synex.
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    Multiplied by TV by DX,
    which is minus sign X.
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    All divided by.
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    V. Squared. Via skull sex so
    squared is called squared X.
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    So we have cost squared X.
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    Minus times minus gives us
    positive sign squared X.
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    Divided by Cos squared X.
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    Now one of our trigonometric
    identity's is called squared X
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    plus sign. Squared X equals 1.
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    So we can substitute 1
    divided by Cos squared X.
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    Which is the same as sex
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    squared X. So our derivative of
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    Tan X. Is sex squared X?
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    Let's have a look at
    another example. This time Y
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    equals sex. And again will
    write sex in terms of sine and
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    cosine, which is one over call
    sex. And again we have our
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    quotient you over V.
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    So I see Y by ZX equals
    V times. Do you buy the X?
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    Minus U times TV by ZX?
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    Or divided by 3 squared.
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    So in this case I you equals 1.
    So I do you buy DX is 0.
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    RV equals
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    Cossacks. So I do feet
    by DX equals minus sign X.
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    So our derivative do why by DX
    is equal to V, so it's cause
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    X. Multiplied by do you buy
    DX which is 0?
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    Minus you choose one.
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    Times DV by The X which is
    minus sign X.
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    All divided by V squared, which
    is called squared X.
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    So we have 0.
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    Minus minus that becomes
    positive. One lot of Cynex
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    so have sign X divided by
    Cos squared X.
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    And we can write this.
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    As. One over Cos
    X multiplied by sign X over
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    call sex. Which
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    is. Sex tonics,
    so a derivative of
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    sex is sex tonics.
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    Let's put those in our table.
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    So our derivative of Tan X.
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    Sex squared X.
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    A derivative of sex.
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    Is sex tonics?
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    And in a similar way are
    derivative of Cortex will be
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    minus cosec squared X.
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    And the derivative of
    cosec X is minus Cosec
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    X Cotex.
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    Now we going to extend this
    further to look at what happens
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    when we have time of MX.
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    And the sack of MX.
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    The cast of air Max and the
    cosec of MX.
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    Let's have a look at the
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    calculation. So this time
    we have Y equals the
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    time of MX.
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    And what we're going to do here
    is to use a substitution.
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    So you were going to substitute
    instead of MX, so RY is
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    equal to tan you.
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    So if we calculate, do you buy
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    DX? We get M.
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    And if we find the derivative of
    Y with respect to you while
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    we've just learned how to do
    that, the derivative of tan you
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    is sex squared you.
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    So I do why by DX.
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    Is equal to do Y fi
    CU multiplied by to you by
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    DMX. The Wi-Fi do you? Is
    sex squared you?
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    Multiplied by do you buy DX,
    which is RM. Let's put the
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    constant first M times this X
    squared. You now we've
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    introduced EU and we don't want
    the you were trying to find the
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    wide by DX we wanted in terms of
    X. So what we do is we
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    substitute for the you so we
    have M times 6 squared of MX.
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    So what we get when we
    differentiate ton of MX?
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    Is the sex squared MX multiplied
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    by M? Let's have a
    look at another example.
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    This time, let's have a look
    at Y equals cosec of MX.
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    Again, we're going to let you
    equal MX as our substitution.
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    And therefore Y is equal to
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    Kosek. Of you.
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    Do you buy the X is equal
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    to end? And our DY bye see you.
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    The derivative of cosec you.
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    Is minus cosec you
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    cut you? We
    know that divide by DX.
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    It cause divide by do you
    multiplied by do you buy DX?
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    RDY by do you?
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    Is minus cosec you caught
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    you? Multiplied by do you buy DX
    which is M?
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    So we have minus N.
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    Cosec you caught you.
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    Now we need to substitute for
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    these use. So we
    have minus M, Cosec
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    MX cult MX.
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    So the derivative of
    cosec MX is minus M.
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    Cosec MX caught MX.
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    Now let's go back to our table
    and put those in.
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    So we found that the
    derivative of tan of MX.
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    Is M6 squared
    MX?
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    The derivative of SAC MX.
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    Is M times the sack of
    MX times a ton of MX?
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    And in a similar way we
    would find that the koptev
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    MX is minus M Times the
    cosec squared MX.
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    And cosec MX. The
    derivative is minus M
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    times cosec MX cult
    MX.
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    OK, we're going to extend our
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    table further. Find a
    clean page.
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    Let's just rewrite our headings
    are function F of X.
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    That derivative
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    DFDX
    Or F Dash Devex.
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    And we're going to
    look now at.
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    The inverse. Sign
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    minus 1X. It's
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    minus 1X. And tonics.
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    Minus 1 - 1 X.
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    And then finally will look at A
    to the power X.
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    Let's do the calculation now for
    sign minus 1X.
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    So this time Y
    equals sine minus 1X.
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    What we're going to do here is
    right this as sign Y equals X.
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    Now we want to find the
    derivative of Y with respect to
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    X. So we want to differentiate.
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    Side Y with respect to X.
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    And differentiate RX with
    respect to X.
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    His right hand side is very easy
    because the derivative of X is
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    just one with respect to X.
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    This side we're going to
    differentiate implicitly.
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    And when we do that, we get the
    derivative of sine. Y is cause Y
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    multiplied by divided by DX.
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    So if we now divide both sides
    by cause why we get DY by DX
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    equals 1 divided by cause Y?
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    Now that's fine, except for the
    fact that we actually wanted in
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    terms of X.
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    So we need to use one of our
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    trigonometric identity's. I'm
    looking to use call squared Y
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    plus sign squared Y equals 1.
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    And if we rearrange this, we
    get cause Y.
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    Equals 1 minus sign squared Y
    and then we want 'cause why not
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    cause squared Y? So we're going
    to square root it.
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    Now we introduce slight problem
    here because when we square root
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    something we could have a
    positive answer. Or we could
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    have a negative answer. So we
    need to just look at that for a
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    moment. Now if we look at the
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    diagram. Of Y equals sine minus
    One X, so here's a graph of Y
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    equals sine minus One X will see
    that there are many values of Y.
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    So our values of X. So for one
    value of X we could have lots of
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    different values of Y.
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    So that's not a function, and
    for it to be a function, we
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    just want one set of values of
    Y for each set of values of X.
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    So what we need to do is
    actually restrict the section
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    that we're looking at, and if
    we restrict it to between minus
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    π by two and plus five by two,
    just this section here.
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    Then we do have a function
    because we have just one value
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    of Y for each value of X.
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    And if we look at this section,
    we can see that the gradient.
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    Is increasing and it's positive,
    so we're actually going to take
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    the positive square root when we
    take the square root.
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    So if we continue down here one
    over cause Y, we can actually
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    write as one over the square
    root of 1 minus sign squared Y.
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    Now we still wanted in terms of
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    X. But if we look here, we
    know that sign Y is equal to
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    X, so we can substitute in
    for sine squared Y here and
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    put X squared. So we get one
    divided by the square root of
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    1 minus X squared.
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    White was cause minus One X can
    be done in a similar way and is
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    left as an exercise. But note
    there that when you look at the
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    square root should have to take
    a negative square root rather
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    than a positive one.
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    Let's have a look now at Y
    equals 10 - 1 X.
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    In the same
    way, we're going
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    to write Tan,
    Y is equal
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    to X. And
    again, take the derivative of
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    each side with respect to
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    X. I can right hand
    side is easy, it's just
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    one. And we differentiate this
    side implicitly, which gives us.
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    The derivative of tan Y is sex
    squared Y multiplied by DY by
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    ZX. Sophie now divide
    both sides by sex squared
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    Y we get 1 / 6
    squared Y.
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    Now, in a similar way to before
    we need to use a trigonometric
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    identity. And we need to use the
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    one. What sex squared Y
    equals 1 plus?
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    10 squared, Y.
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    So if we substitute that.
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    In here we have 1 /
    1 + 10 squared Y.
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    And since 10 Y is X, then that's
    just one over 1 plus X squared.
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    So the derivative of tan minus
    One X is 1 / 1 plus X
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    squared. Let's go back and put
    those in our table now.
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    So the derivative of sine minus
    One X is one over the square
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    root of 1 minus X squared.
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    A derivative of Cos minus One X
    is minus one over the square
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    root of 1 minus X squared and
    our derivative of tan minus One
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    X is one over 1 plus X squared.
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    Let's have a look now at Y
    equals 8 to the power X.
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    Y equals A to the power X and
    this time we're going to start
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    by taking natural logarithms of
    each side. So log to the base E
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    of Y is equal to log to the base
    E of A to the power X.
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    Now we're going to use one of
    the laws of logarithms to bring
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    down the power. So on the right
    hand side we have X times log to
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    the base E of A.
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    Now we want to find the
    derivative of this, so we want
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    to differentiate log to the base
    E of Y with respect to X and we
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    want to differentiate X times
    log to the base E of A with
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    respect to X.
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    Now if we look at the right hand
    side, first log to the base E of
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    a is a constant, so it's a
    constant times X.
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    So when we differentiate with
    respect to X, we just get the
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    constant log to the base E of A.
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    If we differentiate log to the
    base E of Y with respect to X.
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    We get one over Y multiplied by
    DY by DX.
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    So do why by DX is going to
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    be equal. To now, to get this
    side is being divided by Y, so
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    want to multiply both sides by
    Y. So we've got Y multiplied by
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    log to the base E of A.
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    But we wanted in terms of X, not
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    Y. But we know that Y
    equals A to the power X,
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    so I do why by DX becomes
    A to the power X
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    multiplied by log to the
    base E of A.
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    Now it's interesting to note
    here that actually if the
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    constant a is equal to.
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    2.7 one and so on. That's
    the value of E.
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    Then what we have is Y equals
    E to the power X and out
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    the why by DX is equal to.
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    A to the power X? Well, that's
    E. To the power X multiplied by
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    log to the base E of a, which in
    this case is E and log to the
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    base E of E is one, so we get.
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    Our derivative is E to the power
    X, which is what we expected.
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    Let's go and complete our table
    now, then with the derivative of
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    A to the X as A to the
    power X multiplied by log to the
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    base E of A.
Title:
www.mathcentre.ac.uk/.../8.9a%20Use%20table%20derivs%2012%20functs.mp4
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