If ` log _(10)((x^(3)-y^(3))/(x^(3)+y^(3))) = 2`, then show that ` (dy)/(dx) = (-99x^(2))/(101y^(2))`

Explanation

Given, `log_(10)(x^(3)-y^(3))/(x^(3)+y^(3)) = 2`
Or `log_(10)(x^(3)-y^(3)) - log_(10)(x^(3)+y^(3))=2`
Now, differentiating with respect to x, we have ,
` 1/(x^(3)-y^(3)) (3x^(2)-3y^(2)(dy)/(dx)) - 1/(x^(3)+y^(3))(3x^(2)+3y^(2)(dy)/(dx))`
or `(3x^(2))/(x^(3)-y^(3))-(3y^(2))/(x^(3)-y^(3))(dy)/(dx)-(3x^(2))/(x^(3)+y^(3)) - (3y^(2))/(x^(3)+y^(3))(dy)/(dx)=0`
or ` - (3y^(2))/(x^(3)-y^(3))(dy)/(dx)-(3y^(2))/(x^(3)+y^(3))(dy)/(dx) = - (3x^(2))/(x^(3)-y^(3))+(3x^(2))/(x^(3)+y^(3))`
or ` -3y^(2)(dy)/(dx)[1/(x^(3)-y^(3))+1/(x^(3)+y^(3))]`
`=-3x^(2)[1/(x^(3)-y^(3))-1/(x^(3)+y^(3))]`
or `y^(2)(dy)/(dx) [1/(x^(3)-y^(3))-1/(x^(3)+y^(3))]`
` = x^(2)[1/(x^(3)-y^(3))-1/(x^(3)+y^(3))]`
or `y^(2)(dy)/(dx)[(x^(3)+y^(3)+x^(3)-y^(3))/((x^(3)-y^(3))(x^(3)+y^(3)))]`
` = x^(2)[(x^(3)+y^(3)-x^(3)+y^(3))/((x^(3)-y^(3))(x^(3)+y^(3)))]`
or ` y^(2)(dy)/(dx) [(2x^(3))/((x^(3)-y^(3))(x^(3)+y^(3)))]`
` = x^(2)[(2y^(3))/((x^(3)-y^(3))(x^(3)+y^(3)))]`
or ` x(dy)/(dx) = y`
or ` (dy)/(dx) = y/ x ` ... (i)
Also, ` log_(10) . (x^(3)-y^(3))/(x^(3)+y^(3)) = 2 `
or ` (x^(3)-y^(3))/(x^(3)+y^(3)) = 10^(2)`
or `(x^(3)-y^(3))/(x^(3)+y^(3)) = 100`
or ` x^(3) - y^(3) = 100 (x^(3)+y^(3))`
or ` x^(3)-y^(3) = 100 x^(3) + 100 y^(3)`
or ` -99 x^(3) = 101 y^(3)`
or ` y^(3)/x^(3) = - 99/101`
or ` y/x= - (99x^(2))/(101 y^(2))` ...(ii)
From (i) and (ii) , we get,
` (dy)/(dx) = - (99x^(2))/(101 y^(2))`