## 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
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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
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Y equals TAN X.
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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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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
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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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