Question at position 1 If 7x2+4y2=17x^2+4y^2=1, then dydx=\frac{dy}{dx}=7x+4y7x+4y4y7x\frac{4y}{7x}14x+8y14x+8y1−14x8y\frac{1-14x}{8y}−7x4y-\frac{7x}{4y}

Question

Question at position 1 If 7x2+4y2=17x^2+4y^2=1, then dydx=\frac{dy}{dx}=7x+4y7x+4y4y7x\frac{4y}{7x}14x+8y14x+8y1−14x8y\frac{1-14x}{8y}−7x4y-\frac{7x}{4y}

BlackTom AI · Accepted Answer

Question: If 7x^2 + 4y^2 = 1, then dy/dx = ?

Option 1: 7x + 4y  Very wrong form. This looks like a linear expression mix of variables, not a derivative. Differentiating implicitly would involve y' and a ratio, not a simple sum of 7x and 4y.

Option 2: 4y/7x This reverses the ratio and also omits the negative sign that typically appears when solving for y'. It would imply a slope opposite to what the actual implicit differentiation yields, and also ignores the fact that y' multiplies y to balance the x-term, not a simple division of y by x.

Option 3: 14x + 8y This is just the result of differentiating both terms with respect to x as if y were constant; it ignores that y is a function of x. In implicit differentiation, you must include y' when differentiating y^2 terms, so this cannot be correct.

Option 4: 1 − 14x / 8y This mixes a leftover constant-like term with a fraction of x and y in a way that doesn’t arise from the standard implicit differentiation of 7x^2 + 4y^2 = 1. It misplaces the balance of terms and the negative relationship between x and y that appears in the derivative.

Option 5: − 7x / 4y This is the correct structure. Differentiating 7x^2 + 4y^2 = 1 with respect to x gives 14x + 8y(y') = 0. Solving for y' yields y' = dy/dx = −(14x)/(8y) = −(7x)/(4y). The negative sign reflects the inverse relationship between x and y on the implicit curve, and the denominator 4y comes from the derivative of 4y^2.

Summary: Only Option 5 matches the correct implicit-differentiation result, while the others fail to account for y being a function of x, the correct algebraic rearrangement, or the sign. The key steps are differentiating, collecting terms with y', and solving 14x + 8y y' = 0 for y'.

Question at position 1 If 7x2+4y2=17x^2+4y^2=1, then dydx=\frac{dy}{dx}=7x+4y7x+4y4y7x\frac{4y}{7x}14x+8y14x+8y1−14x8y\frac{1-14x}{8y}−7x4y-\frac{7x}{4y}单项选择题

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