Question

If `sqrt3(x^2 + y^2) = 4xy`, then find `(dy)/(dx)  at (1/2, sqrt3/2)`.

Answer 1

Given: `sqrt3(x^2 + y^2) = 4xy`

Differentiate implicitly:

`sqrt3(2x + 2y(dy)/(dx)) = 4(y + x(dy)/(dx))`

Simplify:

`2sqrt3x + 2sqrt3y(dy)/(dx) = 4y + 4x(dy)/(dx)`

Rearrange:

`(2sqrt3y - 4x)(dy)/(dx) = 4y - 2sqrt3x`

`2sqrt3y(dy)/(dx) - 4x(dy)/(dx) = 4y - 2sqrt3x`

`(dy)/(dx)(2sqrt3y - 4x) = (4y - 2sqrt3x)`

`(dy)/(dx) = (4y - 2sqrt3x)/(2sqrt3y - 4x)`

Substitute `x = 1/2, y = sqrt3/2`

Numerator:

= `4 * sqrt3/2 - 2sqrt3 * 1/2`

= `2sqrt3 - sqrt3`

= `sqrt3`

Denominator:

= `2sqrt3 * sqrt3/2 - 4 * 1/2`

= 3 − 2

= 1

∴ `(dy)/(dx) = sqrt3`