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

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

BlackTom AI · Accepted Answer

We start by restating the problem in our own words and listing the answer choices clearly.
Question: If 7x^2 + 4y^2 = 1, Find dy/dx.
Answer options:
- 7x + 4y
- 4y / 7x
- − 7x / 4y
- (1 − 14x) / (8y)
- 14x + 8y

Now, analyze each option step by step.
Option 1: 7x + 4y. This is a linear expression in x and y and has no relation to a slope dy/dx arising from implicit differentiation of 7x^2 + 4y^2 = 1. It ignores the necessary quotient or ratio that typically appears when solving for dy/dx from an equation involving x and y. Therefore, this cannot be correct.
Option 2: 4y / 7x. This is a ratio of y and x, but the derivative from differentiating 7x^2 + 4y^2 = 1 with respect to x yields a relationship between dy/dx, x, and y. If we try to isolate dy/dx naively as 4y/(7x), we would have transformed the equation improperly; the correct method should involve dy/dx multiplied by the derivative of y with respect to x, not just a simple ratio of y and x. Consequently, this is not correct.
Option 3: − 7x / 4y. This option matches the standard implicit differentiation result. Differentiating both sides of 7x^2 + 4y^2 = 1 with respect to x gives 14x + 8y(dy/dx) = 0. Solving for dy/dx yields dy/dx = −14x/(8y) = −(7x)/(4y). This aligns precisely with the form in this option, making it correct.
Option 4: (1 − 14x) / (8y). This expression resembles a rearranged form with an added 1 in the numerator, which does not come from the straightforward differentiation and solving process. The presence of 1 in the numerator indicates a misstep in isolating dy/dx; the correct numerator should be proportional to x without an extra constant term. Therefore, this is incorrect.
Option 5: 14x + 8y. This is a linear combination of x and y, not a slope. It does not express dy/dx and does not reflect the required ratio after solving the implicit differentiation equation. Hence, this cannot be correct.
In summary, the only option that faithfully represents the result of implicit differentiation for 7x^2 + 4y^2 = 1 is the ratio with a negative sign, i.e., −7x/4y. The other choices either ignore the implicit relationship, mis-handle the algebra, or misrepresent the derivative as a simple linear combination instead of a ratio. Remember, the correct procedure is: differentiate to get 14x + 8y(dy/dx) = 0, then solve for dy/dx to obtain dy/dx = −14x/(8y) = −(7x)/(4y).

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

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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}

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