Question

A curve is represented by the equation 2y3 + xy - 6x + 3 = 0. Part A: Determine dy dx (10 points) Part B: Evaluate dy at the point (1, 1). (10 points) dx Part C: Determine d°у (10 points) dx2 day Part D: Evaluate at the point (1, 1). (10 points) dx2

Answer 1

Part A

`\[ (dy)/(dx) = -((2y^3 + 6))/((6 + x)) \]`

Part B:

`\[ (dy)/(dx) \Big|_((1, 1)) = -((2(1)^3 + 6))/((6 + 1)) = -(8)/(7) \]`

Part C:

`\[ (d^2y)/(dx^2) = (24y^2)/((6 + x)^2) \]`

Part D:

`\[ (d^2y)/(dx^2) \Big|_((1, 1)) = (24(1)^2)/((6 + 1)^2) = (24)/(49) \]`

Step-by-step explanation:

In Part A, we find the first derivative
`\((dy)/(dx)\`) using implicit differentiation on the given equation. The result is
`\(-((2y^3 + 6))/((6 + x))\).`

For Part B, we substitute the point (1, 1) into the expression for
`\((dy)/(dx)\)` to find the slope of the curve at that point. The calculation yields
`\(-(8)/(7)\).`

Moving on to Part C, the second derivative
`\((d^2y)/(dx^2)\)` is found by differentiating
`\((dy)/(dx)\)` with respect to x. The result is
`\((24y^2)/((6 + x)^2)\).`

Lastly, in Part D, we evaluate the second derivative at the given point (1, 1), resulting in
`\((24)/(49)\).`

These calculations provide a comprehensive understanding of the curve's behavior and its concavity at the specified point. The negative value in Part B indicates a decreasing slope, and the positive value in Part D suggests a concave-up curvature at the point (1, 1).

Answer 2

To find dy/dx, differentiate both sides of the equation using implicit differentiation and solve for dy/dx.

Step-by-step explanation:

In order to find the derivative dy/dx of the curve represented by the equation 2y³ + xy - 6x + 3 = 0, we can apply implicit differentiation. First, differentiate both sides of the equation with respect to x. The derivative of 2y³ with respect to x is 6y²(dy/dx), the derivative of xy with respect to x is y + x(dy/dx), and the derivative of -6x + 3 with respect to x is -6. Setting the equation to zero and solving for dy/dx, we get:

6y²(dy/dx) + y + x(dy/dx) - 6 = 0

(6y² + x)(dy/dx) = 6 - y

dy/dx = (6 - y) / (6y² + x)