BAM! Mr. Tarrou. In this video we are going to find the coordinates of the
points on the curve Y to the fourth power plus four times X times y to the
third power minus X to the fourth is equal to 16 at which the derivative the
slope of the tangent line at that point the instantaneous rate of change
whatever you want to call it is equal to zero or you know what are the point of
tangency such that the slope of the tangent line is equal to zero many ways
you can say this well how are we going to take the derivative of this equation
and set it equal to zero or find out where it's equal to zero are we going to
do it explicitly are we going to solve it for y in terms of x say so that we
have a clean dy/dx and then take the derivative I don't think we're gonna
want to solve that for y and possibly not even probably not even be able to
and be much simpler just to simply do implicit differentiation so that's what
we're either learning or reviewing in this lesson so we're going to take the
derivative of both sides of this equation with respect to X as this
notation indicates and pay attention to the chain rule as we complete this
process so we have Y to the fourth power okay well we're just going to use the
power rule we're going to bring that 4 down reduce the power by 1 and then
remember we're not taking a derivative with respect to Y we're taking the
derivative with respect to X we have to finish by multiplying by the derivative
of what's inside that function that original power of 4 will that's dy/dx
now + and you can leave the 4 in there or pull it out from and leave the fort
as part of this what this product of two functions we have a 4x times a Y to the
third this second term is the product of two functions so thus when we take the
derivative we have to go through the product rule which is going to be and
there might be some cyber asians because this is multiplication in addition which
functions are commutative but we're going to do the first
time's the derivative of the second so we're looking at a parentheses first
function for X the derivative of the second function well the derivative of Y
cubed is x 3y bring that power down by one squared and then finish the chain
rule dy/dx or I'm gonna write Y prime on the next line now we're going to then
add that with the we just did the first times the derivative derivative of the
second function or factor so now we're going to do plus the second factor y
cubed times the derivative of the first and it's the derivative of 4x so the
derivative of 4x oh or think of it as four times X to the first we're going to
bring that power down by one and say that we have four times X to the zero
power which is just you know one and just for teaching purposes thinking
about finishing that chain rule for every single step of the derivative
because I don't know if this is a review or the introduction to implicit
differentiation but we're going to finish that chain rule we had that four
times X to the zero and then that times DX DX taking the derivative of what's
inside that power original power of 1 with respect to X well that's going to
just cancel out so the derivative of 4x with respect to X is just 4 that
hopefully didn't just bore you trying to over explain that okay so anyway minus
the derivative of X to the fourth is negative because it's minus in the first
place for X to the third again finishing that chain rule one more time for
teaching purposes DX over DX which is going to again just simply cancel out in
the derivative of 16 the derivative of a constant regardless of you know what
variable you're taking it with respect to the derivative with respect to is
going to you know be 0 so we have 4y cubed Y prime plus we have 4 times 3 is
12 X Y squared Y Prime here we've got plus 4y cubed minus 4x
cubed is equal to 0 ok so now what do we have we've got our two terms here that
both have a Y Prime in it that's we're solving for the Y prime or dy/dx so
we're going to move these terms of 4y cubed minus 4x cubed over to the other
side with addition or subtraction so we have 4x cubed minus 4y cubed we're going
to then take with these two terms they both have a common factor of Y prime and
they also have some other common factors bool we'll take care of that in a second
so we're going to take that Y prime out of each of these two terms we're trying
to find that dy/dx isolate that Y prime so we're going to divide both sides of
this equation by this factor and I'm also going to do some some factoring
here we have a common factor of 4 and both of these terms here we both of
these terms have a common factor of 4 and a common factor of Y squared so
we're going to have 4y squared 4y cubed divided by 4 y squared is just
going to be Y and 12 divided by 4 is 3 we've just taken out the Y squared so we
have 3x the question asks us to find the points where the derivative is equal to
0 and I've taken up the whole board up to clean this up in a second
I don't necessarily care where the denominator is equal to 0 because that's
where the derivative is undefined it would help us find locations or points
where the tangent line is vertical but I do want to pay attention to the
numerator so they're just going through this work here we have Y prime is equal
to this is a difference of cubes the
pattern for this is X cubed minus y cubed is going to be a factor of X minus
y and then multiply that 2x squared we're going to change this sign and
we're going to multiply these terms of x and y so we have x squared plus XY plus
y squared over 4 y squared times y plus 3x okay so that is our first derivative
that is what we're trying to make sure or set equal to zero so let me get this
out of my way and look what we have going on here in the numerator we have a
factor which is simply the wave form that's been always before we have X
minus y so when is X minus y going to be equal to zero that's and this is what
it's making the challenge of increasing the difficulty or the challenge of this
question it's set up so that you're not going to find you know an immediate x
value here that's why this is a little bit of an interesting question this is
going to be equal to zero when X is equal to Y okay we'll use that in a
moment what about this other factor where is x squared plus XY plus y
squared equal to zero well really both of these equations have two unknowns and
they're kind of independent of each other this factor doesn't need to equal
zero at the same time this factor is equal to zero is there a relationship
between x and y like we have here where they're equal that the subtraction is
zero is there a relationship between x and y such that this is equal to zero
well if we take this if we take this in the idea that we're seeing this as a
quadratic we could say well a is equal to one on the yellow and green
looks very similar we have here we could set the B is equal to Y and C is equal
to Y squared and see if we can't you know let the quadratic formula tell us
if there's a way of setting up x and y so that such that it comes out to be 0
you might be able to see that this also you may intuitively know that it's ever
gonna be true but we're gonna say here X is equal to think of like a you know ax
squared plus BX or B X plus C is equal to 0 your quadratic so X is equal to the
opposite of B plus or minus the square root of b squared minus 4 ay C all over
2a then you see what's going to happen here we have negative Y that's is equal
to negative Y plus or minus the square root of negative 1 minus 4 is negative 3
y squared over 2 well that's just there's no solution there it's y squared
is always going to be a positive value regardless of what Y is equal to you
take a number you square you have a positive answer and that multiplied by
negative 3 means that you're taking the even root
square root of a negative number which is gonna give you imaginary solutions
now you could say well why did you bother doing all that we could say that
while this didn't work x squared plus y squared excuse me x squared plus XY plus
y squared is equal to 0 when both x and y equals 0 okay so maybe this was
unnecessary maybe we have a couple of points like maybe 0 0 is a coordinate of
a point 0 0 is a point where this function has a
slope of zero but is the solutions of zero and zero for x and y which again
are what needs to happen for this third factor in our numerator to be zero is it
a solution to this equation I don't think so zero to the fourth power plus 4
times 0 times 0 to the third power minus 0 to the fourth power it is definitely
not going to be equal to 16 so this is just simply a no-go now we do know that
from this factor the derivative is equal to 0 when X is equal to Y so let's see
what we get there if X is equal to Y ok so we have Y to the 4th plus 4 X which
we're now going to do per doing substitution here we're gonna I'm going
to plug in Y everywhere that I see in X so we have a 4y Y cubed minus I naught 4
to the fourth is equal to 16 that is going to give us 1/4 plus 4y to the
fourth minus y to the fourth is equal to 16 well 1 minus 1 is 0 we have 4 y to
the fourth is equal to 16 Y to the fourth is equal to 4 now Y is equal to
want to raise both sides of this to a power of 1/4 right to make that power
equal to 1 power 2 power 4 times 1/4 is 1 and I don't you can write the fourth
root of 4 but 4 is 2 squared right so now you're looking at two to the two
over fourth power or two to the 1/2 power which is the square root of two
okay now hold on a second we just took up fourth roots an even root which means
that there's a possibility of having both a positive and a negative answer so
come back here plus or minus so Y can take on the values of plus or minus
the square root of 2 and since we know that the only way for the derivative to
equal zero is if x and y are the same values then our points the coordinates
of the points where this curve has a instantaneous rate of change those
derivative slope of zero is at da da da and green for some reason the square
root of 2 square root of 2 and negative square root of 2 negative square root of
2 I'm miss a true pam go to your homework
you