Implicat differntiation

Question

Daughter was working the problem and got stuck with the last part. Any help is appreciated.

Find the point on the curve where 16y^3(8xy-3) / (4xy-1)^3 = 0

Thanks in advance.

Answer 1

I'm not sure I understand the problem "find the point on the curve" where the convoluted fraction = 0

that fraction = 0 when the numerator = 0

the numerator = 0 when any of its factors =0

set each factor in the numerator = 0, ignore the denominator

16y^3(8xy-3)=0

y=0 or xy=3/8

(x,0) will be a point satisfying the original fraction =0, where x is any real number

that's the x axis

also xy=3/8 is a rectangular hyperbola, with points such as (1, 3/8) and (3/8, 1)

making the original fraction equal zero

that hyperbola and the x axis include all points that make the fraction = 0

there is an infinite number of such points

But then maybe you meant something different as the problem,

since you have it listed under the category implicit differentiation and calculus

as if you want to take the derivative of the fraction.

if that's what you want, use the quotient rule, but it gets very tedious

denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all over the denominator squared. Hard to believe they'd give a problem that tedious

also the "curve" may be referring to another equation? maybe there's more information?

"got stuck on the last part" suggests there are other parts that may be useful or necessary to solve the last part?

and you want "the point" suggesting there is only one such point, which is not the solution above

sometimes the problem is figuring out what the problem is