To differentiate the equation y = 3y^3 + 4x^2 + 2x + 2xy + 1 with respect to x, we need to apply implicit differentiation since y is expressed in terms of x.
Let’s differentiate each term individually:
- For the term 3y^3, using the chain rule, we get:
d/dx(3y^3) = 9y^2(dy/dx) - For 4x^2, it’s straightforward:
d/dx(4x^2) = 8x - For the term 2x:
d/dx(2x) = 2 - For the product 2xy, we need to use the product rule:
d/dx(2xy) = 2(x(dy/dx) + y) - The constant 1 does not change:
d/dx(1) = 0
Putting it all together, we have:
dy/dx = 9y^2(dy/dx) + 8x + 2 + 2(x(dy/dx) + y)
Now, we can rearrange to solve for dy/dx:
dy/dx - 9y^2(dy/dx) - 2x(dy/dx) = 8x + 2 + 2y
Factor out dy/dx:
dy/dx(1 - 9y^2 - 2x) = 8x + 2 + 2y
Finally, we can solve for dy/dx:
dy/dx = (8x + 2 + 2y) / (1 - 9y^2 - 2x)
This is the expression for the derivative of y with respect to x.