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}

查看解析

登录即可查看完整答案

类似问题

At the given point, find the slope of the curve or the line that is tangent to the curve, as requested. 3x2y - π cos y = 4π, slope at (1, π)

Find dy/dx by implicit differentiation. x1/3 - y1/3 = 1

At the given point, find the line that is normal to the curve at the given point. 6x2y - π cos y = 7π, normal at (1, π)

Use implicit differentiation to find the slope of the tangent line to the curve defined by x y 4 + 6 x y = 5 at the point ( 5 , 1 ) .

Use implicit differentiation to find the slope of the tangent line to the curve defined by x y 2 + 2 x y = 5 at the point ( 3 , 1 ) .

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

If 𝑔 ( 𝑥 ) + 𝑥 cos ⁡ ( 𝑔 ( 𝑥 ) ) = 𝑥 9 , find 𝑔 ′ ( 0 ) .

If f ( x ) + x 5 ( f ( x ) ) 2 = 34 and f ( 1 ) = 2 {"version":"1.1","math":"f(x)+x^5\left(f(x)\right)^2=34\text{ and }f(1)=2"} then f ′ ( 1 ) = {"version":"1.1","math":"f'(1)="}

更多留学生实用工具

考试浏览器助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单