Final answer:
To find y' at the point (1,1), differentiate the equation (xy³ + 2x²y = 3) implicitly with respect to x. Substitute x=1 and y=1, then solve for dy/dx to get the value of y' at the point (1,1).
Step-by-step explanation:
To find y' at the point (1,1), we need to differentiate the equation (xy³ + 2x²y = 3) implicitly with respect to x. Let's differentiate both sides of the equation:
d/dx(xy³) + d/dx(2x²y) = d/dx(3)
Using the product rule and chain rule, we get:
y³ + 3xy²(dy/dx) + 4xy(dy/dx) = 0
We can simplify this to:
3xy²(dy/dx) + 4xy(dy/dx) = -y³
Now, let's substitute x=1 and y=1, since we want to find y' at the point (1,1):
3(1)(1²)(dy/dx) + 4(1)(1)(dy/dx) = -(1)³
Simplifying further, we have:
3(dy/dx) + 4(dy/dx) = -1
Combining like terms:
7(dy/dx) = -1
Finally, solving for dy/dx:
dy/dx = -1/7