use implicit differentiation to find dy/dx.
xy^2+4y^3=x-2y
x=-4y^3-2y/y^2-1
xy^2+4y^3=x-2y
x(2y)dy/x + y^2 + 12y^2 dy/dx = 1 - 2dy/dx
dy/x(2xy + 12y^2 + 2) = 1 - y^2
dy/dx = (1 - y^2)/(2xy + 12y^2 + 2)
To find dy/dx using implicit differentiation, follow these steps:
Step 1: Differentiate both sides of the equation with respect to x.
Keep in mind that when differentiating y with respect to x, you need to apply the chain rule.
Differentiating the left side:
d/dx (xy^2 + 4y^3) = d/dx (x) - d/dx (2y)
To differentiate xy^2 with respect to x, apply the product rule:
(x)(d/dx(y^2)) + (y^2)(d/dx(x))
To differentiate 4y^3 with respect to x, apply the chain rule:
4(y^3)(d/dx(y))
The right side differentiates to:
d/dx (x - 2y) = 1 - 2(d/dx (y))
Combining all the terms, we have:
(x)(d/dx(y^2)) + (y^2)(d/dx(x)) + 4(y^3)(d/dx(y)) = 1 - 2(d/dx (y))
Step 2: Simplify the equation by combining similar terms on each side.
Rewrite the equation as:
(x)(d/dx(y^2)) + (y^2)(d/dx(x)) + 4(y^3)(d/dx(y)) + 2(d/dx(y)) = 1
Step 3: Solve the equation for d/dx(y).
Rearrange the equation to isolate d/dx(y) on one side:
(x)(d/dx(y^2)) + (y^2)(d/dx(x)) + 4(y^3 + 1)(d/dx(y)) = 1
Subtract (y^2)(d/dx(x)) from both sides:
(x)(d/dx(y^2)) + 4(y^3 + 1)(d/dx(y)) = 1 - (y^2)(d/dx(x))
Divide both sides by (y^3 + 1) to isolate (d/dx(y)):
(x)(d/dx(y^2))/(y^3 + 1) + 4(d/dx(y)) = (1 - (y^2)(d/dx(x)))/(y^3 + 1)
Step 4: Solve for dy/dx by simplifying and factoring.
The term on the left, (d/dx(y^2))/(y^3 + 1), simplifies to (2y)/(y^3 + 1) by applying the power rule.
The term on the right, (1 - (y^2)(d/dx(x)))/(y^3 + 1), simplifies to (1 - (y^2)(1))/(y^3 + 1) = (1 - y^2)/(y^3 + 1) by substituting d/dx(x) with 1.
Substitute these results back into the equation:
(x)(2y)/(y^3 + 1) + 4(d/dx(y)) = (1 - y^2)/(y^3 + 1)
Finally, solve the equation for 4(d/dx(y)):
4(d/dx(y)) = (1 - y^2)/(y^3 + 1) - (x)(2y)/(y^3 + 1)
Divide both sides by 4:
d/dx(y) = (1 - y^2)/(4(y^3 + 1)) - (x)(2y)/(4(y^3 + 1))
Therefore, the derivative dy/dx is given by:
dy/dx = (1 - y^2)/(4(y^3 + 1)) - (x)(2y)/(4(y^3 + 1))