Since we know the definition of
the derivative at a given point,
let us consider some examples
of the calculation of this derivatives
by its very definition.
So let us start with, for example,
function x squared in arbitrary point a. Firstly,
let us revisit the definition of the derivative itself.
The derivative at point a, allow function,
is the limit of the relation between the change of
the function at this very point
and the change of the argument at this very point.
If our x approaches the function we are talking of.
So what we're going to do here,
we're going to do the following;
we're going to substitute all the things we know about
our function in our definition
and try to find this limit.
Just pretty simple, pretty straightforward.
So what we do know.
Firstly, we do know that we start with ideas that
we are looking for the derivative at the point a,
then we are looking at the limit at the point a.
That's pretty much the same as the definition.
Then we're going to substitute our function always well
x-squared and then the value
of the function at the point a is a squared.
Once again one of the most simple things there is.
So what we're going to do.
We're going to once again use
our simple school trick since
we are looking at the difference of two squares,
then you can factor denominator
as x minus a multiply it by x plus a.
As a result of division,
both denominator and nominator by x minus a.
We move from the case when we're
looking at the indeterminate form
where denominator approaches 0
and denominator approaches 0.
For the case when we're looking at
limits of the punches simple function;
x plus a, x approaches a, a approaches a.
That's basically as this sum approaches 2a.
That's the idea you actually familiar with from
the very famous relations that
the derivative of x squared is two x.
That's the same as that we got here.
As another example, let us consider function sine of x.
Once again in their arbitrary point,
just to make sure that we all
understand how to calculate it.
So once again let us just roll with
the very definition here,
and to do it we are gong to write that we are
looking at the limit at the point a
of sine of x minus sine of a divided by x minus a.
After doing that, we're going to use
Howard Turgeon and metric.
Knowledge here and substitute
denominator by the multiplication of
two of sine of half the difference
multiplied by cosine of half a sum.
That still doesn't solve the determinant form
here because x minus a
approaches zeros as we are having sine of 0 in
denominator as we still haven't won zero divided by zero.
So what are we going to do,
we are going to move our multiplication by
two to the denominator in the following fashion;
we are going to write it as sine
of half the difference multiply it
by cosine of half a sum,
divided by x minus a divided by two.
So what we are looking at;
we are looking at sine of sum sin,
an apple divided by apple,
where our apple approaches zero since x
approaches a is then x minus
a divided by two approaches zero.
So basically we're looking at
our second important limits right from
our seen about it as limits of
single-valued functions as as
this relation altogether approaches one,
or another case we are just looking at the limit of
cosine of x plus a divided by
two cosine of sum
multiplied by one which are going to meet,
and as a result we get cosine of a.
Once again the famous relation
here is the derivative of
sine function is cosine function.
Okay, so now we know how to
calculate two derivatives of two basic functions.
As you might know,
the table of the derivative of
elementary function is well known and normally we
do not use it to compute once again with definition.
We just roll with the table.
So you should always remember basic derivatives.
For example, derivative of
the power which is always power
multiply it by x powered n minus one.
Then we need to write for
example, derivative of logarithm,
which is one divided by x the derivative of
exponent which is exponent itself.
The derivative of a formula with
different value which is
multiplied by the logarithm of this very value.
Then we get, for example,
our trigonometric derivatives,
we've already established them.
The derivative of cosine is minus sine,
and you also should never forget about
the derivatives of in
the Ross to the symmetric functions.
I'm going to, for example, I'll
always arc tangent function
inverse tangent function or
the derivative of tangent function.
I'm going to use two annotations here.
This is one divided one by one plus x squared.
As a whole table or as a derivative of
elementary functions can be
found in additional materials,
and you are supposed to know by heart through our course.
It's not a mandatory thing but it's kind of will happen
whilst you are tackling
all the tests you're going to look at.
Also,you will be given non-mandatory tests here just
to drill all legal knowledge
about the taking of derivatives.