we will find the derivative of 1/x^2 by using the definition of derivative
prime of a by using the definition of derivative i will use the second derivation of derivative
in other words we have to find the limit as h goes to zero f of a plus h minus f of a or over h
so we have to figure out what the f of a plus h is first so let's do the outside we know that
the function f of x is equal to x to negative two power but let's look at the negative two power s
one over x to the power of u two power and to get f of a plus h i just need to plug in
a plus h into this x right here in another word we have one over a plus h and then square that
a plus h times a plus h on the denominator we end up with a squared plus 2 a h plus h squared
and with that we can get what so let me write this down this is the limit as h goes to zero for this
part f of a plus h this right here is that so i'm just going to bring this down right here
one over a squared plus 2 a h plus h squared and then for the second part
we are going to subtract f of a to get f of a i just need to plug in a into
this x in another word we have 1 over a squared and on the denominator we still have the h
so that's the setup as we can see we have a complex fraction because we have a small
fraction here another one and then a big one to take care of this situation we are going to
multiply the top and bottom by the lowest common denominator of the small fractions namely we are
going to multiply the top and bottom by a squared times that a squared plus 2ah plus h squared
okay multiply this amount and also do the bottom a squared times a squared plus 2ah plus h squared
to continue with this all we need to do is take this and then distribute it into the
top so let's put on the result right here and leave some space let me write down the limit as
h goes to zero so on the top when you take this multiply with that this part that part cancels we
just have a squared left so let me write down a squared and then we are going to subtract
subtract one over a squared times this and the a squared will cancel with the a squared
and make sure that you use the parentheses right here i'm going to subtract parentheses a squared
plus 2ah and then plus h squared on the top that's where the small fractions are and then
once you multiply this out we don't have the small fractions anymore this is the place that we're
going to fix on the denominator let's now multiply out h and that okay so i'll just bring this down
h and then we also have a squared and then this part which is a squared plus 2ah plus h squared
be careful even though this parenthesis and not parenthesis are the same but we cannot cancel
because on the top we still have to combine terms here we have a squared
minus a squared and so on so on right so if you distribute the negative two parenthesis
a squared minus a squared they will cancel each other out when you distribute the
negative into the parenthesis now we are going to end up with negative 2ah
okay negative 2h and we also take the negative multiply with h squared and we get minus h
squared and on the top as we can see we have the h right here
another h right here so i can factor the h out so i'll get the h out and then the parentheses
which i will have negative 2a right here minus another h and that's what we have
this h multiplies this parenthesis and i have the h right here that can cancel out with this atrial
problem and finally see something that cancels out on the top and on the bottom so plugging
zero into all the h that we see on the top we have negative two a that's all we have left minus zero
over a squared on the bottom and then here we have a squared plus 2a and the h is zero
and then plus h squared and once again plugging 0 into h so we have the expression looks like
this what we are going to get is on the top just negative 2a and on the bottom
we have a square this right here is zero this right here is also zero a squared times a squared
we are going to get a to the fourth power on the denominator and then we have a right here
a to the fourth on the bottom so we can cancel out one of the a and finally what we're going to get
is so this will cancel with that we have a to a circle on the bottom the answer will be negative 2
over a to the third power