Find the first partial derivatives of the function. f(x, y) = x⁹y

Solution:

∂f(xy)/∂x and ∂f(xy)/∂y are the first partial derivatives of the given function.

∂f(xy)/∂x = ∂x⁹y/∂x

= y∂x⁹/∂x + x⁹∂y/∂x

While differentiating partially with respect to x, y becomes a constant and therefore,

∂f(xy)/∂x = y(9)x⁸ + ( x⁹)(0)

= 9yx⁸ + 0

= 9yx⁸

∂f(xy)/∂y = ∂x⁹y/∂y

= x⁹∂y/∂y + y∂x⁹/∂y

Since partial differentiation is w.r.t. y then x becomes a constant

∂f(xy)/∂y = x⁹ + y(0) = x⁹

Find the first partial derivatives of the function. f(x, y) = x⁹y

Summary:

The first partial derivatives of the function. f(x, y) = x⁹y are ∂f(xy)/∂x = 9yx⁸ and ∂f(xy)/∂y = x⁹