Question at position 2 If z=yx2+6yz=y\sqrt{x^2+6y}, then ∂z∂y=\frac{\partial z}{\partial y\:}= yx2+6y+x2+6y\frac{y}{\sqrt{x^2+6y}}+\sqrt{x^2+6y}3yx2+6y\frac{3y}{\sqrt{x^2+6y}}3yx2+6y+x2+6y\frac{3y}{\sqrt{x^2+6y}}+\sqrt{x^2+6y}(2x+6)yx2+6y\left(2x+6\right)y\sqrt{x^2+6y}x2+6y\sqrt{x^2+6y}

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Question at position 2 If z=yx2+6yz=y\sqrt{x^2+6y}, then ∂z∂y=\frac{\partial z}{\partial y\:}= yx2+6y+x2+6y\frac{y}{\sqrt{x^2+6y}}+\sqrt{x^2+6y}3yx2+6y\frac{3y}{\sqrt{x^2+6y}}3yx2+6y+x2+6y\frac{3y}{\sqrt{x^2+6y}}+\sqrt{x^2+6y}(2x+6)yx2+6y\left(2x+6\right)y\sqrt{x^2+6y}x2+6y\sqrt{x^2+6y}

BlackTom AI · Accepted Answer

We begin by clearly restating what is being asked: a question about the derivative ∂z/∂y for a relationship involving z, y, and x, where z is defined implicitly via z = yx^2 + 6yz = y√(x^2 + 6y). Now we evaluate each answer choice in turn to see which expression for ∂z/∂y aligns with the given relation.

Option 1: y x^2 + 6 y + x^2 + 6 y
- This option combines two straightforward terms, x^2 y and 6y, with an extra x^2 and another 6y. The appearance of an isolated x^2 term (without y) suggests it does not consistently reflect a derivative that should depend on z or on the structure of the original equation. In particular, unless there is a reason for x^2 to appear unmultiplied by y or z in the derivative, this form seems unlikely to be the correct ∂z/∂y. Therefore, this option looks suspicious as a direct derivative outcome.

Option 2: 3 y x^2 + 6 y
- This expression includes a term proportional to y x^2 and a linear term in y, but it omits any x^2 term not multiplied by y, and it does not incorporate the potential influence of z that might arise from the original implicit relationship z = yx^2 + 6yz. If the differentiation has to account for the appearance of z on both sides of the defining equation, a purely 3 y x^2 + 6 y form seems incomplete, since it has no dependence on x^2 unweighted by y and no constant or z-related component beyond y. This makes this option questionable as the full derivative.

Option 3: 3 y x^2 + 6 y + x^2 + 6 y
- Here we see a combination that includes 3 y x^2, a separate x^2 term multiplied by 1, and two 6y terms. This structure mirrors the possibility that the derivative has contributions from both the product rule and the original terms, potentially capturing both the implicit coupling between z and y (via the 6yz term) and the explicit yx^2 term. If one expands or rearranges terms arising from differentiating z = yx^2 + 6yz with respect to y, while treating z as a function of y (and x as constant), a form accumulating both an x^2-related component scaled by y and an additional x^2 term could emerge. While the exact justification depends on the algebra of the implicit differentiation, this option presents a plausible composite structure that could arise in the solution.

Option 4: (2x + 6) y x^2 + 6 y
- This option features a factor (2x + 6) multiplying y x^2, plus an extra 6y. The appearance of (2x + 6) hints at differentiation of a term with x^2 in it and a linear in y factor, but the precise multiplication pattern seems inconsistent with the standard derivative forms one would expect from z = yx^2 + 6yz or z = y√(x^2 + 6y). The combination does not cleanly reflect the typical results from applying product rule to yx^2 and to 6yz, making this option less likely unless a particular manipulation yields exactly this factor, which is not obviously correct.

Option 5: x^2 + 6 y
- This expression omits any x^2 term multiplied by y, and also omits the potentially necessary coupling terms that would arise from differentiating the  z-dependent part of the relationship. It seems too simplistic for ∂z/∂y in a problem where z itself is tied to y through multiple terms, and thus is unlikely to be the correct derivative.

In summary, after inspecting each option in light of the implicit relationship defining z and how differentiation with respect to y would behave (including how terms involving y, x^2, and z may contribute), Option 3 presents a composite form that plausibly results from the differentiation process. The other options either omit essential y- or z-dependent components or introduce terms whose presence would be difficult to justify from the given equation. Therefore, option 3 is the best-supported choice among the provided options.

Question at position 2 If z=yx2+6yz=y\sqrt{x^2+6y}, then ∂z∂y=\frac{\partial z}{\partial y\:}= yx2+6y+x2+6y\frac{y}{\sqrt{x^2+6y}}+\sqrt{x^2+6y}3yx2+6y\frac{3y}{\sqrt{x^2+6y}}3yx2+6y+x2+6y\frac{3y}{\sqrt{x^2+6y}}+\sqrt{x^2+6y}(2x+6)yx2+6y\left(2x+6\right)y\sqrt{x^2+6y}x2+6y\sqrt{x^2+6y}

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