just because i'm a calculus teacher a lot of people like to ask me when is the best time to
take the derivative well let's take a look right now it's 3:20 pm and here we go that's the answer
and we will be differentiating the product of these two integral functions
and as i told you we will make derivatives cool this year so here we go this is the product so of
course use the product rule i'm going to keep the first function so we will have the integral going
from 1 to x squared and then we have 2t over 1 plus t squared dt and then we will have to
differentiate the second one and to differentiate an integral function when you have a number
to a function all we have to do is plug in this function into the t and use the chandu from here
so we will get 1 over 1 plus replace the t with l and x and then we still have the parenthesis
to the second power and then use the chandu multiplied by the derivative of that which is
1 over x and then we can continue we are going to add the second function which is the integral
going from 1 to l and x and then we have 1 over 1 plus t squared dt and then again same thing
we are going to just plug in the x square into the t and also here so we are going to get
2 times x squared over 1 plus x squared and then square and then we have to multiply by the
derivative of x squared which is 2x so when you do this this is technically called the fundamental
theorem of calculus part one along with the chen lu all right so that's how we differentiate
integral functions yeah i know you guys want me to actually figure this out and figure this
out i will don't worry all right so let's do this in our head to integrate this notice that
the derivative of the bottom is precisely 2t so this is just integrating one over u right in the
u world after you do the u substitution letting u equal the bottom so we will just get ln of 1 plus
t squared like so and notice i didn't put on the absolute value because everything says positive
and then we just have to go from 1 to x squared and check this out this part which was how to
plug in and plug in subtract so we will get ln of 1 plus x to the second and then to the second
so that's to the fourth power and then minus plugging one here work that out you get one two
so this is really that part in fact if earlier we were going to figure this out first and then do
the derivative we will still have the same answer and that's why we don't need to worry about the
number check this out the integral is this if we differentiate this function the derivative of ln
of one plus x to the fourth power is one over x to the fourth one plus x to the fourth power
and then multiply by the derivative of this which is four x raised to the power we do have four x to
the third power all right and then the derivative of the minus ln2 is just 0 that's why we didn't
need to worry about the number all right and then perhaps we clean this up a little bit
how though i don't know let me just perhaps i'll just put him down here then i guess yeah so that's
the first part and let's go ahead and do the same thing for the second part all right do this in
your head let u equal this which is just going to be integrating 1 over u squared and if you do that
you get negative 1 over u so this part is negative 1 over 1 plus t all right so just integrate that
and then just plugging plugging one and l and x so we are going to get let's see
plugging lnx first so we get okay and we have some denominators so i'm just going right here
so we have negative 1 over 1 plus ln x and then minus and then minus so become plus and put a
1 here so we have one half so work that out you will get that and then the rest and again if you
look at this and do the derivative you will end up with this and your window to this yeah but anyway
uh work this out we get 4x to the third power over 1 plus x to the fourth power and
yeah i'm just gonna stop right here why not huh
nothing else that we can do double check triple check um
let me see
you you can try combine the fractions get the common denominators but we all
don't know it's not necessary unless you have to find the
maximum of this function so maybe you will have to set equal to zero in that case maybe you have to
combine all the fractions and set the numerator equal to zero but i'm gonna do that to you
guys alright anyway that's what i have for you guys today what do you guys have for me though
what shall we differentiate next leave a comment down below let me know that's it