Second Order Derivatives Video Lecture | Mathematics (Maths) Class 12 - JEE

in this video we're going to look at the
second order derivative so so far we
have looked at how to find d-y-by-dx and
Awards the first derivative
whenever you differentiate again you get
d 2 y over DX squared this is called the
second derivative and also you will have
seen F of our F dashed X like so for the
first derivative when you're going to be
differentiate that again you'd get f
dash dash or F Double Dash X like that
ok this and this is the second order
derivative that's what you get one of
you differentiate a function and
differentiate it again ok let's look at
an example so here we've got y equals 4x
cubed plus 5 and the question asks to
find a dy by DX and B D 2 y over DX
squared so first of all let's set off
with a so a has found with dy by DX is
so first of all we know y equals 4x
cubed plus 5
well let's differentiate this with dy by
D X would equal bringing the 3 down 12 x
squared and differentiating 5 gives you
zero so the answer would be 12x squared
so dy by DX is 12x squared next we want
to find the d2 y over DX squared so we
want to differentiate again so d2 y over
DX squared would equal when you
differentiate this you'll bring the 2
down
so 24x and we're just apart by one
what's 1 so the answer would be 24x so
to get d 2 y over DX squared you
differentiate and you differentiate
again or you just differentiate your dy
by DX next this time we're given the y
equals 5 x squared plus 5 over x squared
I'm weepin asked to find dy by DX and
then D 2 y by DX squared so let's first
of all start off with a so y equals 5 x
squared plus 5 over
x squared first of all before we
differentiate this we want to get the
center format that's easily
differentiable differentiable it's
easily he can easily differentiate so
let's look at this x squared on the
denominator and let's bring it up so
we're going to get y equals 5x squared
plus bringing this up would be 5x to the
minus 2 now we can differentiate so d2
dy by DX equals bring the power down 10
X to the power of 1 bring the power down
so 5 times minus 2 is minus 10x whenever
you just reduce the power by 1 minus 2
take where 1 will be minus 3 so dy by DX
is 10x minus 10x to the power of minus 3
Part B is we need to differentiate again
find D 2 y by DX squared so D 2 y over
DX squared equals differentiate in 10x
would be 10 bring in the power down
minus 10 times minus 3 would be plus 30x
and reducing the power by 1 which would
be to the power of minus 4 so d 2 y over
DX squared would be 10 plus 30x to the
power of minus 4 remember we could
rearrange before these back into a
format which you may want to substitute
values into so here we had 10x minus
another instead of writing minus 10x to
the minus 3 we could bring the X to the
minus 3 to the denominator so you get 10
over X square cubed and likewise here
you can have 10 plus 30 and you could
bring the X to the 4 to the Dominator to
give you X to the 4 like that okay so
that's it