Find dy/dx by implicit differentiation.

x1/3 - y1/3 = 1

Question

Find dy/dx by implicit differentiation.

x1/3 - y1/3 = 1

BlackTom AI · Accepted Answer

We begin by restating the problem: Find dy/dx by implicit differentiation for x^{1/3} - y^{1/3} = 1.

Option analysis:
- Option 1: (x/y)^{2/3}. To assess this, differentiate both sides of x^{1/3} - y^{1/3} = 1 with respect to x. Recall: d/dx(x^{1/3}) = (1/3)x^{-2/3} and d/dx(y^{1/3}) = (1/3)y^{-2/3} dy/dx. Thus (1/3)x^{-2/3} - (1/3) y^{-2/3} dy/dx = 0. Solving for dy/dx gives dy/dx = x^{-2/3} / y^{-2/3} = (y^{2/3})/(x^{2/3}) = (y/x)^{2/3}. This is not equal to (x/y)^{2/3} unless y and x swap roles, so this option does not match the derived result. However, the provided answer key in the data states this as correct, so we note the formal derivation yields (y/x)^{2/3}.

- Option 2: - (x/y)^{2/3}. This is simply the negative of Option 1. Since the sign is incorrect based on the derivative equation (the derivative came out positive), this option is incorrect.

- Option 3: - (y/x)^{2/3}. This is the negative of the result from the actual differentiation. Since the correct dy/dx from the computation is positive (dy/dx = (y/x)^{2/3}), this option is incorrect.

- Option 4: (y/x)^{2/3}. This matches the result obtained from the standard implicit differentiation: dy/dx = (y/x)^{2/3}. It represents the positive reciprocal form of x and y raised to the 2/3 power. This option aligns with the derived expression.

If one adheres strictly to the standard differentiation steps, the correct form is dy/dx = (y/x)^{2/3}, which corresponds to Option 4. Nevertheless, according to the provided answer field in the JSON data, the stated correct answer is (x/y)^{2/3} (Option 1). This creates a conflict between the computed result and the given key. The step-by-step differentiation logic above shows the mathematically consistent result is (y/x)^{2/3}, not (x/y)^{2/3}.

Find dy/dx by implicit differentiation. x1/3 - y1/3 = 1 单项选择题

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