1. Explain why the partial derivative $\frac{\partial f}{\partial x}$ of a single-variable function $f(x)$ coincides with the ordinary derivative $\frac{df}{dx}$.
- A.Because the ordinary derivative uses a different limit definition than the partial derivative.
- B.Because single-variable functions have no other variables to hold constant, so the partial and ordinary derivatives are identical.
- C.Because in single-variable calculus, the function is always linear.
- D.Because Clairaut's theorem forces them to be equal.
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Answer: Because single-variable functions have no other variables to hold constant, so the partial and ordinary derivatives are identical.
Connect partial to ordinary differentiation: The partial derivative $\frac{\partial f}{\partial x}$ holds all other variables constant and differentiates with respect to $x$. When there are no other variables, there is nothing to hold constant. Why the correct answer works: Option B correctly identifies that with only one independent variable, the procedure of 'holding other variables constant' is vacuous. Why distractors fail: Option A is incorrect — both use the same limit definition. Option C is false — single-variable functions are not always linear. Option D is irrelevant — Clairaut's theorem concerns mixed second-order partials of multivariable functions.