Question

Let x2 + 4y2 = 16. Which expression represents StartFraction d Superscript 2 y Baseline Over d x squared EndFraction?

StartStartFraction negative y + x StartFraction d y Over d x EndFraction OverOver 4 y squared EndEndFraction

StartStartFraction negative y + 2 x StartFraction d y Over d x EndFraction OverOver 2 y cubed EndEndFraction

StartStartFraction negative y minus (8 minus x) StartFraction d y Over d x EndFraction OverOver 4 y squared EndEndFraction

StartStartFraction negative y minus 2 (8 minus x) StartFraction d y Over d x EndFraction OverOver 2 y cubed EndEndFraction

Answer 1

To find the second derivative d²y / dx² of the given equation, implicit differentiation is used twice. After the first differentiation, we solve for dy/dx, and then differentiate that result with respect to x to find the second derivative.

Step-by-step explanation:

The student is asking for the second derivative of the equation x2 + 4y2 = 16 with respect to x, also denoted as d2y / dx2. To find this, we must first find the first derivative dy/dx by implicit differentiation and then differentiate once more to obtain the second derivative.

First, implicitly differentiate both sides of the equation with respect to x:

2x + 8y(dy/dx) = 0

Now we solve for dy/dx:

dy/dx = -x / (4y)

Next, we differentiate dy/dx implicitly with respect to x to find the second derivative:

d2y / dx2 = d/dx (-x / (4y)) = d/dx (-1/4(y)) * x + (-x) * d/dx (-1/4(y))

After substituting and simplifying, the correct expression for the second derivative is StartStartFraction negative y minus 2 (8 minus x) StartFraction d y Over d x EndFraction OverOver 2 y cubed EndEndFraction.