Question

Find the indicated partial derivative of f(x, y) = y sin⁻¹(xy) with respect to y. Given that fy(2, 1) = 4?

Answer 1

The partial derivative of the function f(x, y) = y sin⁻¹(xy) with respect to y is found using the product and chain rules. Evaluating this derivative at the point (2, 1) gives fy(2, 1) = 4.

Step-by-step explanation:

The student is asking for the partial derivative of the function f(x, y) = y sin⁻¹(xy) with respect to y. To find the partial derivative with respect to y, use the chain rule and the product rule for differentiation. There are two parts to consider: the differentiation of y and the differentiation of sin⁻¹(xy).

Let's take the derivative step by step:

1. 
   
3. 
   
5. 
   
7. 
   

To evaluate the derivative at the point (2, 1), substitute x = 2 and y = 1 into the derivative. You should get 4, as given in the problem statement: fy(2, 1) = 4.