Partial derivatives

Question

Consider the following.

f(x, y) = x²y − 3xy²

(b) Find f(3, y)

 and use it to calculate f_y(3, y).

Answer 1

To find f(3, y), we need to substitute x = 3 in the expression for f(x, y):

g(y)= f(3, y) = (3)^2 y - 3(3)y^2 = 9y - 9y^2

To find fy(3, y), we need to take the derivative of g(y) with respect to y:

g'(y)= fy(3, y) = (3)^2 - 6(3)y = 9 - 18y

Therefore, the function f(3, y) = 9y - 9y^2 and fy(3, y) = 9 - 18y.