Final answer:
To find dy/dx by implicit differentiation for the equation 3x² - 5xy - y² = 3, apply the chain rule and product rule accordingly and solve for dy/dx, resulting in dy/dx = (5y - 6x) / (5x + 2y).
Step-by-step explanation:
To find dy/dx by implicit differentiation for the equation 3x² - 5xy - y² = 3, we first differentiate both sides of the equation with respect to x. Remember that y is a function of x, so when differentiating terms with y, we use the chain rule.
Differentiating 3x² with respect to x gives 6x. For the term -5xy, we apply the product rule: the derivative of the first function (-5y) times the second function (x) plus the first function times the derivative of the second function (-5x(dy/dx)). For -y², the chain rule gives us -2y(dy/dx).
Now we combine these results:
6x - 5y - 5x(dy/dx) - 2y(dy/dx) = 0
Next, we solve for dy/dx:
(5x + 2y)(dy/dx) = 5y - 6x
So, dy/dx = (5y - 6x) / (5x + 2y).