Partial Derivatives - Differentiation, Business Mathematics & Statistics Video Lecture | Business Mathematics and Statistics - B Com

so let's say I have some multi variable
function like f of XY so they'll have a
two variable input is equal to I don't
know x squared times y plus sine of Y so
a lot of put just a single number it's a
scalar valued function question is how
do we take the derivative of an
expression like this and there's a
certain method called a partial
derivative which is very similar to
ordinary derivatives and I kind of want
to show how there secretly the same
thing so to do that let me just remind
ourselves of how how we interpret the
notation for ordinary derivatives so if
you have something like f of X is equal
to x squared and let's say you want to
take its derivative and I'll use the
liveness notation here DF DX and let's
evaluate it at 2 let's say I really like
this notation because it's suggestive of
what's going on if we sketch out a graph
so you know this axis represents our
output this will be here represents our
input and x squared has a certain
parabolic shape to it something like
that then we go to the input x equals 1
to this little DX here I like to
interpret as just a little nudge in the
X direction and it's kind of the size of
that nudge and then DF DF is the
resulting change in the output after you
make that initial little nudge so it's
this resulting change and when you're
thinking in terms of graphs this is
slope you kind of have this rise over
run for your ratio between the tiny
change of the output that's caused by a
tiny change in the input and of course
this is dependent on where you start
over here we have x equals 2 but you
could also think about this without
graphs if you really wanted to you might
just think about you know your input
space is just a number line and your
output space also is just a number line
the output of F over here and really
you're just thinking of somehow mapping
numbers from here onto the second line
and in that case your initial nudge your
initial little DX would be some nudge on
that number line and you're wondering
how that influences the the function
itself so maybe that causes a nudge
that's you know four times as big and
that would mean your derivative is four
at that point
the reason that I'm talking about this
is because over in the multi variable
world we can pretty much do the same
thing you know you could write DF DX and
interpret that is saying hey how does a
tiny change in the input in the X
Direction influence the output but this
time the way that you might visualize it
you'd be thinking of your input space
here all I'll draw it down here is the
XY plane so this time this is not going
to be graphing the function this is
every point on the plane is an input and
let's say you were evaluating this at a
point like 1/2 okay in that case so
you'd go over to the input that's 1 and
then 2 and then you'd say okay so this
tiny nudge in the input this tiny change
DX how does that influence the output
and in this case the output I mean it's
still just a number so maybe we go off
to the side here and we draw just like a
number line as our output and somehow
we're thinking about the function as
mapping points on the plane to the
number line so you'd say okay that's
your that's your DX how much does it
change the output and you know maybe
this time it changes it negatively
depends on your function and that would
be your DF and you can also do this with
the Y variable right there's no reason
that you can't say DF dy and evaluate at
that same point 1/2 and interpret
totally the same way except this time
your dy would be a change in the Y
Direction so maybe I should really
emphasize here that that DX that DX is a
change in the X Direction here and that
dy is a change in the Y direction and
maybe when you change your F according
to Y it does something different right
maybe you know the output increases and
it increases by a lot it's more
sensitive to Y again it depends on the
function and I'll show you how you can
compute something like this in just a
moment here but first there's kind of an
annoying thing associated with partial
derivatives where we don't write them
with DS in DX DF people came up with
this new notation mostly just to
emphasize to the reader of your equation
that it's a multivariable function
involved
what you do is you say you write a D but
it's got kind of a curl at the top it's
this new symbol and people will often
read it as partial so you might really
like partial F partial Y if you're
wondering by the way why we call these
partial derivatives it's sort of like
this doesn't tell the full story of how
f changes because it only cares about
the X direction neither does this this
only cares about the y direction so each
one is only a small part of the story
so let's actually evaluate something
like this I'm going to go ahead and
clear the board over here I think the
one dimensional analogy is something you
probably have already so ah little
remnants so if you're actually
evaluating something like this here I'll
write it again up here partial
derivative of F with respect to X and
we're doing it at 1/2 it only cares
about movement in the X direction so
it's treating Y is a constant it doesn't
even care about the fact that Y changes
as far as its concerned Y is always
equal to 2 so we can just plug that in
ahead of time so I'm going to say
partial partial X this is another way
you might write it put the expression in
here and I'll say x squared but instead
of writing Y I'm just going to plug in
that constant ahead of time because when
you're only moving in the X direction
this is kind of how how the
multivariable function sees the world
and I'll just keep a little note that
we're evaluating this whole thing at x
equals 1 and here this is actually just
an ordinary derivative right this this
is an expression that's an X you're
asking how it changes as you shift
around X and you know how to do this
this is just taking the route of taking
the derivative the derivative of x
squared times 2 is going to be 4x
because x squared goes to 2 X and then
the derivative of a constant sine of 2
is just a constant is 0 and of course
we're evaluating this at x equals 1 so
your overall answer is going to be 4 and
just for practice let's also do that
with the derivative with respect to Y so
we look over here I'm going to write the
same thing you're taking the partial
derivative of F with respect to Y we're
evaluating it at the same point 1 2 this
time it doesn't care about movement in
the X direction
so as far as its concerned that X just
stays constant at 1 so we'd write 1
squared times y plus sine of Y sine of Y
and you're saying oh I'm keeping track
of this at y equals 2 so that's kind of
you're evaluating at y equals 2 when you
take the derivative this is just 1 times
y so the derivative is 1 this over here
the derivative is cosine cosine of Y
again we're evaluating this whole thing
at y equals 2 so your overall answer it
would be you know 1 plus cosine of 2 I'm
not sure I'm not sure what the value of
cosine of 2 is off the top of my head
but that would be your answer and this
is this is a partial derivative at a
point but a lot of times you're not
asked to just compute it at a point what
you want is a general formula that tells
you hey plug in any point XY and it
should spit out the answer so let me
just kind of go over how you would do
that it's actually very similar but this
time instead of plugging in the constant
ahead of time we just have to pretend
that it's a constant so let me let me
make a little bit of space for ourselves
here really we don't need any of this
anymore I'm going to leave the partial
partial F partial partial Y we want this
as a more general function of X and why
well we kind of do the same thing we're
going to say that this is you know
derivative with respect to X and I'm
using partials just to kind of emphasize
that it's a partial derivative but now
we'd write x squared and then kind of
emphasize that it's a constant value of
y plus the sine and again I'll say Y and
here I'm writing the variable Y but we
have to pretend like it's a constant now
you're pretending that you plug in 2 or
something like that
and you still just take the derivative
so in this case the derivative of x
squared times a constant is just 2 x
times that constant 2x times that
constant and over here the derivative of
a constant is always 0 so that's just
always going to be 0
so this is your partial derivative as a
more general formula if you plugged in
one to two this you'd get what we had
before and similarly if you're doing
this with F of a partial F partial Y we
write down all of the same things now
you're taking it with respect to Y and
I'm just going to copy this formula here
actually
but this time we're considering all of
the exes to be constants so in this case
when you take the derivative with
respect to Y of some kind of constant
you know constant squared is a constant
times y it's just going to equal that
constant so this is going to be x
squared and over here you're taking the
derivative of sine of Y there's no X's
in there so that remains the sine of Y
if you're in clear and now this is a
more general formula if you plugged in 1
2 you would get 1 so outside that's
cosine of Y cosine of Y because we're
taking a derivative so if you plugged in
1 2 you know you would get 1 plus the
cosine of 1 which is what we had before
so this this is really what you'll see
for how to compute a partial derivative
you pretend that one of the variables is
constant and you take an ordinary
derivative and then you back of your
mind you're thinking this is because
you're just moving in one direction for
the input and you're saying how that
influences things and then you know you
might move in one direction for another
input and see how that influences things
in the next video I'll show you what
this means in terms of graphs and slopes
but it's important to understand the
graphs and slopes are not the only way
to understand derivatives because as
soon as you start thinking about vector
valued functions or functions with
inputs of higher dimensions than just 2
you can no longer think in terms of
graphs but this idea of nudging the
input in some direction seeing how that
influences the output and then taking
the ratio you know the ratio of that
output nudge to the input nudge that's a
more general way of viewing things and
that's going to be very helpful moving
forward in multi variable cap