Final answer:
To find d2y/dx2 in terms of x and y for the equation x²y² - 16x = 3, differentiate the equation twice with respect to x. Simplify the resulting expression to find the second derivative.
Step-by-step explanation:
To find d2y/dx2 in terms of x and y for the equation x²y² - 16x = 3, we need to differentiate the equation twice with respect to x. Let's start by differentiating both sides of the equation:
d/dx(x²y² - 16x) = d/dx(3)
Using the power rule and the chain rule, we get:
2x(y²) + x²(2y)(dy/dx) - 16 = 0
Simplifying this expression gives:
2xy² + 2x²y(dy/dx) - 16 = 0
Now, differentiating again with respect to x:
d/dx(2xy² + 2x²y(dy/dx) - 16) = d/dx(0)
Using the power rule, the product rule, and the chain rule, we get:
2y² + 4xy(dy/dx) + 2x²(dy/dx)² + 2x²(d²y/dx²) - 0 = 0
Simplifying this expression gives the second derivative:
2y² + 4xy(dy/dx) + 2x²(dy/dx)² + 2x²(d²y/dx²) = 0