Vector Calculus -- chain rule question I have a simple chain rule homework problem. I feel like im missing something here. The statement of the problem: Determine $$\frac{\partial z}{\partial x} \text{ when } z = xy f \left( \frac{x}{y} \right).$$ My attempt: $$\frac{\partial z}{\partial x} = y f \left( \frac{x}{y} \right) + xf' \left( \frac{x}{y} \right).$$ just by product rule and simple chain rule from single variable calculus. I'm kind of weirded out by this, i feel like im overlooking something. The book's notation (Marsden Vector Calculus 5th ed) is much more complicated looking than what I'm doing here. Am I fudging the bucket? If I am what have I missed conceptually?

Your answer appears to be correct. To follow up and make your answer less ambiguous. You should use $\frac{\partial f}{\partial x}(\frac{x}{y})$ because f is a function of more than one variable.