find the dy/dx for:
y^2=(2x^3)y+5x
With y considered a function of x, differentiate both sides of the equation with respect to x. Then solve for dy/dx.
2y dy/dx = (2x^3) dy/dx + y*(6x^2)+ 5
dy/dx(2y-2x^3) = 6x^2*y +5
dy/dx = [6x^2*y+5)/(2y-2x^3)
2y(dy/dx) = 2x^3(dy/dx) + (6x^2)y + 5
2y(dy/dx) - 2x^3(dy/dx) = (6x^2)y + 5
dy/dx(2y - 2x^3) = 6x^2y + 5
dy/dx = ((6x^2)y + 5)/(2y - 2x^3)
To find the derivative of y with respect to x, dy/dx, for the given equation y^2 = (2x^3)y + 5x, we can use implicit differentiation.
Step 1: Differentiate both sides of the equation with respect to x.
d/dx(y^2) = d/dx((2x^3)y + 5x)
Step 2: Apply the chain rule on the left side of the equation.
2y * dy/dx = d/dx(2x^3)y + d/dx(5x)
Step 3: Simplify each term on the right side of the equation.
For the term d/dx(2x^3)y, we have:
d/dx(2x^3)y = 2xy * dy/dx + (2x^3) * dy/dx
For the term d/dx(5x), we have:
d/dx(5x) = 5
Now, our equation becomes:
2y * dy/dx = 2xy * dy/dx + (2x^3) * dy/dx + 5
Step 4: Rearrange the equation to isolate dy/dx.
Group the terms that contain dy/dx on one side:
2y * dy/dx - 2xy * dy/dx - (2x^3) * dy/dx = 5
Factor out dy/dx:
dy/dx * (2y - 2xy - 2x^3) = 5
Divide both sides of the equation by (2y - 2xy - 2x^3):
dy/dx = 5 / (2y - 2xy - 2x^3)
Therefore, the derivative dy/dx for the given equation is 5 / (2y - 2xy - 2x^3).