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.