
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 WiFi 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.