Find dy/dx for the following:
1) x^3 + 3x^2*y + y^3 = 4
I get -3(x+2y)/x^2 + y^2
it's wrong though.
2) x + y^2= xy^2
I get y^2 + 1/ 2y- 2xy
this is also wrong.
can you explain thanks
x^3 + 3x^2*y + y^3 = 4 --------->
d[x^3 + 3x^2*y + y^3] = d(4) = 0 --->
3 x^2 dx + 6xy dy + 3x^2 dy + 3y^2 dy = 0
--->
3 x^2 + 6xy dy/dx + 3x^2 dy/dx + 3y^2 dy/dx = 0
(6 xy + 3 x^2 + 3 y^2) dy/dx = -3 x^2
dy/dx = - x^2/(2 xy + x^2 + y^2) =
-[x/(x+y)]^2
Hmmm I get
x^3 + 3x^2*y + y^3 = 4
3x^2 + 6xy + 3x^2 y' + 3y^2 y' = 0
y'(3x^2 + 3y^2) = -3x^2-6xy
y' = -x(x+2y)/(x^2+y^2)
you must have lost a factor of 3 somewhere
x + y^2= xy^2
1 + 2yy' = y^2 + 2xyy'
y'(2y-2xy) = y^2-1
y' = (y^2-1) / 2y(1-x)
you must have missed a sign change somewhere
Of course! Let's go through both problems and find the correct derivatives step by step.
1) To find dy/dx for the equation x^3 + 3x^2y + y^3 = 4, we will use implicit differentiation.
Start by differentiating both sides of the equation with respect to x:
d/dx(x^3) + d/dx(3x^2y) + d/dx(y^3) = d/dx(4)
The derivative of x^3 with respect to x is 3x^2.
The derivative of 3x^2y with respect to x involves the product rule. We treat y as a function of x, so the derivative of y is dy/dx.
Therefore, d/dx(3x^2y) = 3x^2 * dy/dx + 2(3xy).
The derivative of y^3 with respect to x is 3y^2 * dy/dx.
The derivative of 4 with respect to x is 0, as it is a constant.
Now we have:
3x^2 + 3x^2 * dy/dx + 6xy + 3y^2 * dy/dx = 0
Rearranging and factoring out dy/dx, we get:
dy/dx = (-3x^2 - 6xy) / (3x^2 + 3y^2)
So the correct derivative is dy/dx = (-3x^2 - 6xy) / (3x^2 + 3y^2).
2) To find dy/dx for the equation x + y^2 = xy^2, we can again use implicit differentiation.
Start by differentiating both sides of the equation with respect to x:
d/dx(x) + d/dx(y^2) = d/dx(xy^2)
The derivative of x with respect to x is simply 1.
The derivative of y^2 with respect to x involves the chain rule. We treat y as a function of x, so the derivative of y^2 is 2y * dy/dx.
The derivative of xy^2 with respect to x involves the product rule. We treat y^2 as a function of x, so the derivative of xy^2 is y^2 * dy/dx + 2xy * dy/dx.
Now we have:
1 + 2y * dy/dx = y^2 * dy/dx + 2xy * dy/dx
Rearranging and factoring out dy/dx, we get:
dy/dx(2y - y^2 - 2xy) = 1
Dividing both sides by (2y - y^2 - 2xy), we obtain the correct derivative:
dy/dx = 1 / (2y - y^2 - 2xy)
So the correct derivative is dy/dx = 1 / (2y - y^2 - 2xy).
I hope this explanation helps! Let me know if you have any further questions.