What are the first partial derivatives of the function f(x, y) = x^9y?

To find the first partial derivatives of the function f(x, y) = x⁹y, we need to differentiate the function with respect to each variable while treating the other variable as a constant.

Partial Derivative with respect to x

The partial derivative of f with respect to x is denoted as f_x. We apply the power rule of differentiation:

f_x =  

∂/∂x (x⁹y) = 9x⁸y

Partial Derivative with respect to y

The partial derivative of f with respect to y is denoted as f_y. Here, since we treat x as a constant, the differentiation is straightforward:

f_y =  

∂/∂y (x⁹y) = x⁹

Conclusion

Thus, the first partial derivatives of the function are:

• f_x = 9x⁸y
• f_y = x⁹

These derivatives represent the rate of change of the function f with respect to each of the variables x and y.