Consider problems 10-11 (13.3) in Larson’s 9e calculus textbook:

A student just beginning to work with partial derivatives might wonder why, in the partial derivative with respect to *x* in problem 10, *y* vanishes, whereas, in problem 11, *y*^{3} remains even when we differentiate with respect to *x*. Here’s the key difference. In the function in problem 11, we are multiplying *x*^{2} by *y*^{3}. In the function in problem 10, we were subtracting the *y* term from the *x* term. In problem 11, think of *y*^{3} as a constant that is multiplying *x*. Imagine that, in place of *y*^{3}, you have simply 7. If you had 7*x*^{2}, your derivative would be 14*x*. In other words, we would multiply by 7; the 7 wouldn’t go anywhere. In problem 11, *y*^{3} is a constant that multiplies *x*, so we must retain it, which is why . If the *y* term were being added to or subtracted from the *x* term, then in the partial derivative with respect to *x*, the *y* term would indeed vanish, as the constants would become 0 when differentiated with respect to *x*. On the other hand, because *y*^{3} as a constant is multiplying the *x* term in problem 11, it will remain as is when differentiating with respect to *x*.

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