1. **Problem:** Find all first partial derivatives of the function $$f(x,y) = (2x - y)^4$$. 2. **Formula and rules:** - The partial derivative with respect to $$x$$ treats $$y$$ as a constant. - The partial derivative with respect to $$y$$ treats $$x$$ as a constant. - Use the chain rule: if $$f = g(h(x,y))$$, then $$f_x = g'(h) \cdot h_x$$ and $$f_y = g'(h) \cdot h_y$$. 3. **Calculate $$f_x$$:** - Let $$u = 2x - y$$, so $$f = u^4$$. - Then $$f_x = 4u^3 \cdot \frac{\partial}{\partial x}(2x - y) = 4(2x - y)^3 \cdot 2 = 8(2x - y)^3$$. 4. **Calculate $$f_y$$:** - Similarly, $$f_y = 4u^3 \cdot \frac{\partial}{\partial y}(2x - y) = 4(2x - y)^3 \cdot (-1) = -4(2x - y)^3$$. 5. **Final answer:** $$ \boxed{f_x = 8(2x - y)^3, \quad f_y = -4(2x - y)^3} $$