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