To find y/dx by implicit differentiation of the equation x² + 4xy + y² = 4, we will differentiate both sides of the equation with respect to x.
Starting with the equation:
x² + 4xy + y² = 4
We differentiate each term:
- For the term x², the derivative is 2x.
- For the term 4xy, we use the product rule. The derivative is 4(y + x(dy/dx)).
- For the term y², the derivative is 2y(dy/dx).
Thus, differentiating both sides gives:
2x + 4(y + x(dy/dx)) + 2y(dy/dx) = 0
Next, we can simplify this equation:
2x + 4y + 4x(dy/dx) + 2y(dy/dx) = 0
Now, we combine the terms that contain dy/dx:
4x(dy/dx) + 2y(dy/dx) = -2x – 4y
Factoring out dy/dx from the left side:
(4x + 2y)(dy/dx) = -2x – 4y
Finally, we can isolate dy/dx:
dy/dx = 1( -2x – 4y) / (4x + 2y)
Therefore, the derivative dy/dx is:
dy/dx = 1( -2x – 4y) / (4x + 2y)
In conclusion, using implicit differentiation, we found that:
dy/dx = (-2x – 4y) / (4x + 2y).