Final answer:
To find the partial derivative of f(x, y) = y sin⁻¹(xy) with respect to y, we use the chain rule. Applying the chain rule, we find fy(x, y) = sin⁻¹(xy) + y * (1/√(1 - (xy)²)) * x. Substituting the given values, fy(4, 1/8) = sin⁻¹(4/8) + (1/8) * (1/√(1 - (4/8)²)) * 4.
Step-by-step explanation:
To find the partial derivative of f(x, y) = y sin⁻¹(xy) with respect to y, we will use the chain rule. The chain rule states that if we have a function g(u) and a composition of functions f(g(x)), then the derivative of f(g(x)) with respect to x is given by f'(g(x)) * g'(x). In this case, g(u) = sin⁻¹(u) and f(g(x)) = y sin⁻¹(xy).
Applying the chain rule, the derivative of f(x, y) with respect to y is: fy(x, y) = sin⁻¹(xy) + y * (1/√(1 - (xy)²)) * x.
Substituting the values x = 4 and y = 1/8 into the equation, we get: fy(4, 1/8) = sin⁻¹(4/8) + (1/8) * (1/√(1 - (4/8)²)) * 4.