Simple derivative of different functions - Differentiation, Business Mathematics & Statistics Video Lecture | Mathematics for Airmen Group X - Airforce X Y / Indian Navy SSR

all right so at this point you've now
seen some of the shortcut rules for
finding derivatives which is going to
make life a lot easier at least when it
comes to finding derivatives so here we
just got three examples and I'm just
going to find the derivative of each one
of these so the first one's quite easy
so we've got f of x equals x to the
tenth so the derivative rule says that
whenever you have a variable to a power
we just pull that number out front so
the 10 is going to come right out front
and then we simply subtract 1 from our
exponent so we would have 10 minus 1 or
9 and that's our derivative it says hey
the derivative is just going to be 10
times X to the 9th so you know if you
tried to use that long definition f of X
plus h minus f of X over H you'd take
you quite a bit of work to get there so
kind of nice how quick and easy we can
get this solution let's look at our next
one here G of x equals x to the 5 halves
plus 3 X plus 4 well ok we'll do the
same thing here ok so the 5 over 2
that's going to come out front
let's write this a little better so our
5 over 2 is going to come right out
front ok then we have to subtract 1 so
maybe I'll write that so we have to do 5
over 2 minus 1 that's going to be our
new exponent plus well to kind of be
really clear here when you're
multiplying by a constant nothing really
happens to the constant we can just take
the derivative of the sort of the
variable part so the 1 would come out
front and then we would subtract 1 from
the exponent and that would give us 0
okay so we'll come back to this in just
a second
the derivative of a constant is just 0
so the plus 4 normally we don't write
plus 0 but I'll write it here just to be
you know kind of hopefully clear about
what's going on
all right well so we've got five over
two times X to the five halves well we
would have to get common denominators so
instead of minus one we could write that
as two over two
well five minus two will give us three
over two
when we take a number and raise it to
the zero power we're going to get just
one so we can just write three times X
to the zero is three times one again the
plus zero we'll leave it off so it says
our derivative is going to be five half
five halves times X raised to the 3 over
2 plus three so what happens is again
you know the five over two just comes
out front like normal you know hey it's
a fraction but if you subtract one
you're just subtracting two over two in
this case which gives us three halves
when you have a number times a variable
to the first basically the variable just
goes away when you take the derivative
and then the constant just goes to zero
so that would be our derivative for Part
B and last but not least let's look at
our last one here so we've got so we had
H of x equals one over the cubed root of
x and maybe just to remind you remember
the so if we have X raised to the M and
then the in througt of that we can write
that as X raised to the M over N so you
can think about this as being X to the
first power underneath so we could
really rewrite this as 1 over X raised
to the 1/3 power well to use that
derivative rule we need sort of the X to
be in the numerator well I can pull the
X upstairs and I just have to change the
sign on the exponent so really H of X I
can rewrite that as just saying it's X
to the negative 1/3 and now I'm in a
good spot where I can take the
derivative so the negative 1/3 comes
right out front
and then we would have negative 1/3 and
we would subtract one but I can write 1
as 3 over 3 to get common denominators
so we would have negative 1 over 3
well negative one minus three will give
us negative four over three and this is
now our derivative it says it would be
negative one third X to the negative
Four Thirds
we could rewrite this as negative 1 over
3x to the positive 4/3 and if we wanted
to we could even put this back in our
radical notation so kind of using this
little rule up here so it says whatever
is in the denominator that's what the
root is so here we would be taking still
a cube root but now instead of just
having X to the first we would have X to
the fourth power so that's how we could
sort of rewrite it all again using
radical notation so all right again just
kind of some basic rules here you'll use
these all the time in calculus certainly
will be some other rules coming along as
well but again certainly I think once
you've done a few of these using the
definition life seems much easier now so
but again now the fun part is they're
going to give you really complicated
functions to take the derivative of so
it can still be a little tricky but
certainly these shortcuts do make life a
lot easier
when taking derivatives