The task in the image involves computing the partial derivatives of the given polynomial function $f(x, y) = 8x^2y^7 + 4x^3y - y^9$ with respect to $x$ and $y$.

Let’s calculate the partial derivatives step by step:

1. Partial Derivative with respect to $x$ ($\frac{\partial f}{\partial x}$):

• For the term $8x^2y^7$: Differentiate with respect to $x$, treating $y^7$ as a constant.
  $\frac{\partial}{\partial x}(8x^2y^7) = 16xy^7$.