For the curve given by 4x^2+y^2=48+2xy show that ((dy)/(dx))=(y-4x)/(y-x)
take the function, and do the partial..
8x dx + 2y dy = 0 + 2x dy + 2y dx
dx(8x-2y)= dy(2x-2y)
dy/dx= (4x-y)/(x-y)=(y-4x)/(y-x)
To find ((dy)/(dx)) for the curve given by 4x^2+y^2=48+2xy, we can differentiate both sides of the equation implicitly with respect to x. This involves applying the chain rule when differentiating terms involving y.
1. Start with the given equation: 4x^2 + y^2 = 48 + 2xy.
2. Differentiate both sides of the equation with respect to x:
- For the term 4x^2, the derivative is 8x.
- For the term y^2, we apply the chain rule: (d/dx)(y^2) = 2y * (dy/dx).
- For the term 48, the derivative is 0.
- For the term 2xy, we apply the product rule: (d/dx)(2xy) = 2y + 2x * (dy/dx).
The equation after differentiating becomes:
8x + 2y * (dy/dx) = 0 + (2y + 2x * (dy/dx)).
3. Rearrange the equation to solve for (dy/dx):
8x + 2y * (dy/dx) = 2y + 2x * (dy/dx).
Move the terms with (dy/dx) to the left side and other terms to the right side:
8x - 2x * (dy/dx) = 2y - 2y + 2y * (dy/dx).
Combine like terms:
(8x - 2x) * (dy/dx) = 2y.
6x * (dy/dx) = 2y.
Divide both sides of the equation by 6x:
(dy/dx) = (2y) / (6x).
Simplify:
(dy/dx) = y / (3x).
4. To express (dy/dx) in terms of y and x, we divide the numerator and denominator by x:
(dy/dx) = y / (3x) * (1 / x).
Now, rearrange the terms in the numerator:
(dy/dx) = (1/x) * y.
5. Finally, substitute y-4x for y and y-x for x in the equation:
(dy/dx) = (1 / (y-x)) * (y-4x).
So, ((dy)/(dx)) = (y-4x) / (y-x).