An asteroid is following an elliptical orbit described by the equation

4
𝑥
2
+1
𝑦
2
=
36

Use implicit differentiation to find an expression for dy/dx and determine the value of dy/dx when y=1 and x is a positive number.

Hint: to find the value of dy/dx you will need to determine the positive x value associated with y=1.

You will likely need a calculator to find the value of x and shouldn't round your value of x before inserting it into your expression for dy/dx (remember you can use the memory in button on a calculator to store a value for use later). BE SURE TO DOUBLE CHECK YOUR ANSWER. It is easy to type things into a calculator incorrectly.

Enter your answer to 2 decimal places.

Question

An asteroid is following an elliptical orbit described by the equation

4
𝑥
2
+1
𝑦
2
=
36

Use implicit differentiation to find an expression for dy/dx and determine the value of dy/dx when y=1 and x is a positive number.

Hint: to find the value of dy/dx you will need to determine the positive x value associated with y=1.

You will likely need a calculator to find the value of x and shouldn't round your value of x before inserting it into your expression for dy/dx (remember you can use the memory in button on a calculator to store a value for use later). BE SURE TO DOUBLE CHECK YOUR ANSWER.  It is easy to type things into a calculator incorrectly.

Enter your answer to 2 decimal places.

BlackTom AI · Accepted Answer

Differentiate implicitly: for 4x^2 + y^2 = 36, differentiate both sides with respect to x. Relevant concepts: implicit differentiation (treat y as a function y(x)), power rule, and solving for dy/dx.

1) Knowledge points
- d/dx[x^2] = 2x, d/dx[y^2] = 2y·(dy/dx) by chain rule.
- Solve algebraically for dy/dx.
- Use the ellipse equation to find the corresponding x when y = 1.

2) Step-by-step differentiation
Differentiate each term:
d/dx(4x^2) + d/dx(y^2) = d/dx(36)
8x + 2y (dy/dx) = 0

Solve for dy/dx:
2y (dy/dx) = -8x
dy/dx = -8x/(2y) = -4x/y

3) Intermediate calculation to find x when y = 1
Use the original ellipse: 4x^2 + y^2 = 36
Plug y = 1: 4x^2 + 1^2 = 36 → 4x^2 = 35 → x^2 = 35/4 → x = +√(35)/2 (positive root requested).

Now substitute into derivative expression:
dy/dx = -4x/y = -4(√(35)/2)/1 = -2√35

Numeric evaluation:
√35 ≈ 5.91608
-2·5.91608 ≈ -11.83216
Rounded to two decimal places: -11.83

4) Why this is correct
The implicit differentiation step correctly applies the chain rule to y^2 and isolates dy/dx algebraically. Substituting the exact positive x-value √35/2 ensures no rounding error before the final numeric round-off.

5) Common incorrect answers
Typical mistakes: forgetting the factor of 2 from d/dx(y^2), losing the negative sign, or using the wrong x value (e.g., taking the negative root). Any of these produce numerically different and incorrect results.

6) Conclusion
The value of dy/dx at y = 1 with x positive is -11.83, which matches the provided answer.

类似问题

In the equation below, 𝐶 is some constant. 17 𝑥 𝑒 𝑦 − 3 𝑦 = 𝐶 + 6 𝑥 Suppose the equation represents a curve passing through ( 2 , − 1 ) . Evaluate 𝑦 ′ at ( 2 , − 1 ) .

In the equation below, 𝐶 is some constant. 13 𝑥 𝑒 𝑦 − 3 𝑦 = 𝐶 + 4 𝑥 Suppose the equation represents a curve passing through ( 2 , − 1 ) . Evaluate 𝑦 ′ at ( 2 , − 1 ) .

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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}14x+8y14x+8y1−14x8y\frac{1-14x}{8y}−7x4y-\frac{7x}{4y}

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