we will take the derivative of the quadratic formula! okay let's do some math for fun and
as we all know this right here is a standard form of a quadratic equation and let me tell
you guys what i did to this on the other day i differentiated but i didn't do it with respect to
x though what i did was i differentiated this with respect to "a" so that's why we'll use the partial
derivative symbol you know some calculate stuff however i did this wrong first let me show you
i thought that because we're in the a world so only a is the variable and
the other ones are just constants so the derivative that i had was
just x squared because differentiating a with respect to a we get one so we have the x square
and the other are just constants so that's it but you know this is wrong because it was too easy
yeah whenever we are doing like a strange question if it's too easy then most likely it's wrong right
before we continue i would like to tell you guys that this video is sponsored by brilliant
if you want to know more math go ahead and check them out but more about them later let's talk
about why this is wrong well from the quadratic equation we actually have the quadratic formula
and that's x equals negative b plus or minus but just to keep our signature let's just focus on the
class version here you can do the whole process later on again with the minus in between okay
but just plus here and then we have the square root b squared minus 4ac all divided by 2a
so you see x is actually a function of a so x is not a constant in this equation
yeah so that's the bizarre part so let's go ahead and do the following we are going to differentiate
the quadratic formula with respect to a and then we'll get answer for that and then we
will go back to the quadratic equation and we will differentiate this by using plus differentiation
the correct way so let's go ahead and get work right here again let me just use the partial
derivative symbol because we have so many letters here okay partial x with respect to a here we need
to use the quotient rule because we have the a on the bottom also a here so let's go ahead and open
a big fraction bar and then we will have 2a being squared on the bottom and then we put this on the
top first 2a times the derivative of the top derivative of negative b is zero because we are
in the a world and then we add the derivative of this which is just going to be some number
over 2 square root of the inside which is b squared minus 4ac and don't forget to use
the chandu of course we have to multiply by the derivative inside derivative b
squared with respect to a is 0. derivative of negative 4ac with respect to a is negative 4 c
all right that's the first part and then we will have to minus the top function which is negative b
plus square root of b squared minus 4ac and then multiply by the derivative of the bottom
which is just a 2. so that's what we have and now let's clean things up a little bit
as we can see this 2 and not 2 can solve which is pretty nice and we do have this
little denominator here so let's go ahead and multiply the time and bottom by that
so we will multiply by square root of b squared minus 4ac and of course we'll do the same right
here as well b squared minus 4ac so far so good and now let's see what we will be getting
so this is going to be negative 4c times a so i'll write it as negative 4ac
this and that will cancel and then negative 2 times that it becomes positive we want the 2b
and we will have to multiply this right here so open the square root b squared minus
and then minus 2 times this so we have negative 2 and then the square root cancel
so just open the parentheses b squared minus 4ac
and all divided by this and we can open this for a squared square root of b squared minus 4ac and
in fact we can clean things up a little bit right here we see this is negative 2b squared plus 8 ac
so in fact this and that can be combined so we will do that this is going to give us
perhaps i'll write down this term first negative 2 b squared and then plus 4ac
and then that so plus 2b square root of b squared minus 4ac and all divided by yeah
so if you wonder what's the rate of change of x which does to a it's this
so you can have a quadratic equation right and then you can just adjust the a a little bit and
you can find the rate of change yeah so this right here is what we have and as i promised earlier we
will go back here and do implicit differentiation and show you that this is actually incorrect
remember x is a function of a so here when we have a times x squared we will have to use the product
rule yeah that's the part so i'm going to keep the first function which is a and then we multiply
by the derivative of the second derivative of x squared is 2x but x is a function of a
so we multiply by partial x with respect to a like so and then we add the second function which is
x squared and then we multiply by the derivative of the first which is going to be just a 1. good
next though we differentiate dx with respect to a so we are going to add b and then partial x with
respect to a like this and then the derivative c is zero and that will be just equal to zero
now we can actually solve for partial x with respect to a right here i'm going to move this
to the other side so we have negative x squared and then over we have this
and that on the left hand side that we have to divide so we get 2 ax
plus b so that's what we have if you want to have the answer in the implicit form this is
it much better of course but is this really the same as that did i make a mistake again
well we better check how well we know that x is equal to this
so we're just going to be plugging into x here and also x here and see what happens
check this out 2a 2a cancel each other very nicely next this is negative b likewise this is positive
b they also cancel out this is what we'll do next we're going to open the top and then put things in
the right water have a look on the bottom we have this but also this 2a square we will also put on
the bottom which is 4 a squared and then along with that square root of b squared minus 4ac
good here we have a negative just put it in the front and then we are going to square this
negative b squared just b squared here and then we have we'll have to do 2 times this and that
and because this is a minus so it's a minus 2 and then the b and then we have just that part square
root of b squared minus 4ac and lastly we add this square so we add 4 no we add b squared minus 4ac
have a look people b squared plus b squared is
2b squared when we have a negative well because we have to distribute the negative good next you see
that here we have negative times negative so this is indeed the same as that lastly negative times
that so it becomes positive 4ac right there take a look hopefully you liked this video and i would
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i also want to thank you for checking them out and also you can go ahead and try differentiate this
with respect to b and also with respect to c and let me know how it goes