Find the First Partial Derivatives of the Function f(x, y) = x^7y

To find the first partial derivatives of the function f(x, y) = x⁷y, we need to differentiate the function with respect to each variable separately while treating the other variable as a constant.

Partial Derivative with respect to x

The first step is to calculate the partial derivative of f with respect to x, denoted as ∂f/∂x. When differentiating x⁷y with respect to x, we treat y as a constant:

∂f/∂x = 7x6y

Partial Derivative with respect to y

Next, we will find the partial derivative of f with respect to y, denoted as ∂f/∂y. Here, we treat x as a constant:

∂f/∂y = x7

Summary

In summary, the first partial derivatives of the function f(x, y) = x⁷y are:

• ∂f/∂x = 7x⁶y
• ∂f/∂y = x⁷