Derivatives of Composite Functions (Using the Chain Rule) Video Lecture | Mathematics (Maths) Class 12 - JEE

let's say we have the function f of X
which is equal to cosine of X to the
third power which we could also write
like this cosine of X to the third power
and we are interested in figuring out
what f prime of X is going to be equal
to so we want to figure out F prime of X
and as we will see the chain rule is
going to be very useful here and what
I'm going to do is I'm going to first
just apply the chain rule and then maybe
dig into a little bit too to make sure
we draw the connection between what
we're doing here and then what you might
see and maybe some of your calculus
textbooks that explain the chain rule so
if we have a function that is defined as
essentially a composite function notice
this this expression right here we are
taking something to the third power it
isn't just an X that we're taking the
third power we are taking a cosine of X
to the third power so we're taking a
function you could view it this way
we're taking the function cosine of X
and then we're inputting it into another
function that takes it to the third
power so let me put it this way if you
viewed if you say look we could take an
X we put it into one function and that
is that first function is cosine of X so
first we evaluate the cosine and so
that's going to produce cosine of X
cosine of X and then we're going to
input it into a function that just takes
things to the third power so it just
takes things to the third power and so
what are you going to end up with well
you're going to end up with what are you
taking to the third power you're taking
cosine of X cosine of X to the third
power this this is a composite function
you could view this you could view this
as the function let's call this blue one
the function V and let's call this the
function u and so if we're taking X and
into u this is U of X and then if we're
taking u of X into the input or as the
input into the function V then this
output right over here this is going to
be
viii of well what was inputted V of U of
X V of U of X or another way of writing
it I'm going to write it multiple ways
that's the same thing as V of cosine of
X V of cosine of X and so V whatever you
input into it it just takes it to the
third power if you were to write V of X
it would be X to the third power so the
chain rule tells us or the chain rule is
what our brain should say hey it's it
becomes applicable if we're going to
take the derivative of a function that
can be expressed as a composite function
like this so just to be clear we can
write f of X f of X is equal to V of U
of X I know I'm essentially saying the
same thing over and over again but I'm
saying in slightly different ways
because the first time you learned this
it can be a little bit hard to to grok
or really deeply understand so I'm going
to try to write it in different ways and
the chain rule tells us that if you have
a situation like this then the
derivative F prime of X and this is
something that you will see in your
textbooks well this is going to be the
derivative of this whole thing with
respect to U of X so we could write that
as V prime of U of X V prime of U of x
times the derivative of u with respect
to x times u prime of X this right over
here this is one expression of the chain
rule and so how do we evaluate it in
this case let me color code it in a
similar way so the V function this outer
thing that just takes things to the
third power I'll put in blue so f prime
of X another way of expressing it and
I'll use it with more of the
differential notation you could view
this as the derivative of well I'll
write it a couple of different ways you
could view it as the derivative of V the
derivative of V with respect to U and we
get the colors right the derivative of V
with respect to u
that's what this thing is right over
here times the derivative of U with
respect to X so times the derivative of
U with respect to X and just to make
clear so you're familiar with the
different notations you'll see in
different textbooks this is this right
over here just using different notations
and this is this right over here so
let's actually evaluate these things
you're probably tired of just talking in
the abstract so this is going to be
equal to this is going to be equal to
and I'm going to write it out again this
is the derivative instead of just
writing V and u I'm going to write it
let me write it this way this is going
to be I keep wanting using the wrong
colors this is going to be the
derivative of I'm going to leave some
space times the derivative of something
else with respect to something else so
we're going to first take the derivative
of V well V is cosine of X to the third
power cosine of X we're gonna take the
derivative of that with respect to U
which is just cosine of X and we're to
multiply that times the derivative of U
which is cosine of X with respect to X
with respect to X so this one we have
good we've we've seen this before we
know that the derivative with respect to
X of cosine of X cosine use it on that
same color the derivative of cosine of X
well that's equal to negative sine of X
so this one right over here that is
negative sine of X you might be more
familiar with seeing the derivative
operator this way but in theory you
won't see this as often but this helps
my brain really grok what we're doing
we're taking the derivative of cosine of
X with respect to X well that's going to
be negative sine of X well what about
taking the derivative of cosine of X to
the third power with respect to cosine
of X what is this thing over here mean
well if I was taking the derivative if I
was taking the derivative of let me
write it this way if I was taking the
derivative
of X to the third power X to the third
power with respect to X if it was like
that well this is just going to be and
let me put some brackets here to make it
a little bit clearer if I'm taking the
derivative of that that is going to be
that is going to be we bring the
exponent out front it's going to be 3 3
times X 3 times X to the second power 3
times X to the second power so the
general notion here is if I'm taking the
derivative of something whatever this
something happens to be let me just in a
new color it could be I'm doing the
derivative of orange circle to the third
power with respect to orange circle well
that's just going to be 3 times orange
or yellow circle let me make it an
actual orange circle so the derivative
of orange circle to the third power with
respect to orange circle that's going to
be three times the orange circle squared
so if I'm taking the derivative of
cosine of X to the third power with
respect to cosine of X well that's just
going to be this is just going to be
three times cosine of X cosine of X to
the second power to the second power
notice we're just one way to think about
I'm taking the derivative of this
outside function with respect to the
inside so I would do the same thing as
taking the derivative of X to the third
power but instead of an X I have a
cosine of X so instead of it being 3x
squared
it is three cosine of x squared and then
the chain rule says if we want to
finally get the drivet with respect to X
we then take the derivative of cosine of
X with respect to X now that's a big
mouthful but we are at the homestretch
we've then we've now figured out the
derivative it's going to be this times
this so let's see that's going to be
negative three negative three times sine
of X x times cosine squared of X and I
know that was kind of a long way of
saying it I'm trying to explain the
chain rule at the same time but once you
get the hang of it you're just going to
say all right well let me take the dirt
negative of the the outside of something
to the third power with respect to the
inside
let me just treat that cosine of X like
as if it was an X well that's going to
be if I did that that's going to be
three cosine squared of X so that's that
part and that part and then let me take
the derivative of the inside with
respect to X well that is negative sine
of X