To differentiate the expression y + 3y + 4x² + 2x + 2xy + 1 with respect to x, we need to apply the rules of differentiation, including the product rule and the chain rule, since the expression contains both x and y.
First, let’s simplify the expression:
- y + 3y can be combined to get 4y.
- Thus, the expression simplifies to 4y + 4x² + 2x + 2xy + 1.
Now, we differentiate each term with respect to x:
- 4y: Since y can be a function of x, we use the chain rule. The derivative is 4 rac{dy}{dx}.
- 4x²: The derivative of 4x² is 8x (using the power rule).
- 2x: The derivative of 2x is 2.
- 2xy: Here, we need to use the product rule: u = 2x and v = y. The product rule says that u’v + uv’. Thus, the derivative is 2y + 2x rac{dy}{dx}.
- 1: The derivative of a constant is 0.
Now we can add all the derivatives together:
4 rac{dy}{dx} + 8x + 2 + 2y + 2x rac{dy}{dx} + 0.
Combine like terms:
(4 + 2x) rac{dy}{dx} + 8x + 2 + 2y.
Thus, the derivative of the expression y + 3y + 4x² + 2x + 2xy + 1 with respect to x is:
(4 + 2x) rac{dy}{dx} + 8x + 2 + 2y.