Find dy/dx.
7xy^3 − x^2y = 6
Newtons method:
7x(3y^2)dy+7y^3dx-2xydx-x^2 dy=0
dy/dx(21xy^2-x^2)=2xy-7y^2
dy/dx= you do the fraction.
To find dy/dx, we need to use the implicit differentiation technique. In implicit differentiation, we treat y as a function of x and differentiate both sides of the equation with respect to x.
Let's start by differentiating both sides of the equation with respect to x:
d/dx [7xy^3 - x^2y] = d/dx [6]
Using the product rule, we differentiate each term on the left side separately:
(7x)(d/dx[y^3]) + (y^3)(d/dx[7x]) - (2xy)(d/dx[x]) - (x^2)(d/dx[y]) = 0
Now, we need to find the derivatives of y^3, x, and y:
The derivative of y^3 can be found using the power rule:
d/dx[y^3] = 3y^2(dy/dx)
The derivative of x with respect to x is simply 1:
d/dx[x] = 1
The derivative of y with respect to x is dy/dx.
Plugging these derivatives back into the equation, we have:
(7x)(3y^2(dy/dx)) + (y^3)(7) - (2xy)(1) - (x^2)(dy/dx) = 0
Next, simplify the equation:
21xy^2(dy/dx) + 7y^3 - 2xy - x^2(dy/dx) = 0
To find dy/dx, we can isolate it on one side of the equation:
21xy^2(dy/dx) - x^2(dy/dx) = 2xy - 7y^3
Factor out dy/dx:
(dy/dx)(21xy^2 - x^2) = 2xy - 7y^3
Now, divide both sides by (21xy^2 - x^2):
dy/dx = (2xy - 7y^3) / (21xy^2 - x^2)
So, the derivative dy/dx is (2xy - 7y^3) / (21xy^2 - x^2).