Final answer:
To find y' for the equation x² * xy * y² = 7 using implicit differentiation, apply the product rule to each product of functions and then solve the resulting equation for y'.
Step-by-step explanation:
To find y' by implicit differentiation for the equation x² * xy * y² = 7, we need to apply the product rule for differentiation multiple times. The product rule states that the derivative of a product of functions is the derivative of the first function times the second function, plus the first function times the derivative of the second function.
Let's differentiate both sides of the equation with respect to x:
- First, we apply the product rule to x² * xy, which gives 2x * xy + x² * (y + x * y').
- Then, we differentiate y², which is 2yy'.
- Now multiply the two results together, again applying the product rule, we get (2x * xy + x² * (y + x * y')) * y² + x² * xy * 2yy' = 0.
Solve this equation for y' to get the derivative of y with respect to x.