If 3x^2 + y^2 = 7 then evaluate the second derivative of y with respect to x when x = 1 and y = 2. Round your answer to 2 decimal places.

3x^2+y^2 = 7

6x + 2yy' = 0
y' = -3x/y

using implicit differentiation again,
6x + 2yy' = 0
6 + 2y'^2 + 2yy" = 0
y" = -(6+2y'^2)/2y
= -(6+2(9x^2/y^2))/(2y)
= -3(3x^2+y^2)/y^3
= -21/y^3

or, using the quotient rule,
y' = -3x/y
y" = ((-3)y - (-3x)(-3x/y))/y^2
= -3(3x^2+y^2)/y^3
= -21/y^3

To evaluate the second derivative of y with respect to x, we need to find the derivative of the first derivative of y with respect to x.

Step 1: Differentiate the given equation with respect to x to find the first derivative of y:

∂/∂x [3x^2 + y^2] = ∂/∂x [7]
⇒ 6x + 2yy' = 0 [using the chain rule]

Step 2: Solve the equation for y':
⇒ 2yy' = -6x
⇒ y' = -3x / y

Step 3: Differentiate the first derivative of y with respect to x to find the second derivative of y:

∂/∂x [-3x / y] = -3/y * ∂/∂x [x]
⇒ -3y' / y = -3/y

Step 4: Plug in the given values x = 1 and y = 2:

Second derivative of y with respect to x = -3/y = -3/2

Therefore, the second derivative of y with respect to x when x = 1 and y = 2 is -1.50 (rounded to 2 decimal places).

To evaluate the second derivative of y with respect to x, we need to find the derivative of y with respect to x twice.

Step 1: Find the first derivative of y with respect to x.
Given that 3x^2 + y^2 = 7, we can implicitly differentiate both sides of the equation with respect to x.

d/dx(3x^2 + y^2) = d/dx(7)

Using the chain rule, the derivative of y^2 with respect to x is 2y * dy/dx.

6x + 2y * dy/dx = 0

Step 2: Solve for dy/dx.
Rearranging the equation, we can isolate dy/dx.

2y * dy/dx = -6x

dy/dx = -6x / (2y)

dy/dx = -3x / y

Step 3: Differentiate dy/dx with respect to x to find the second derivative of y with respect to x.
Differentiating dy/dx = -3x / y with respect to x:

d/dx(dy/dx) = d/dx(-3x / y)

We can use the quotient rule to differentiate the right side of the equation. The quotient rule states that for functions u(x) and v(x), the derivative of u(x)/v(x) is (v(x) * du/dx - u(x) * dv/dx) / v(x)^2.

Using the quotient rule on -3x / y, we have:

(dy/dx) * d^2y/dx^2 = [(y)(-3) - (-3x)(0)] / y^2

=( -3y ) / y^2

= -3/y

Step 4: Evaluate the second derivative when x = 1 and y = 2.
Substituting x = 1 and y = 2 into the equation -3/y, we have:

-3/2

Therefore, the second derivative of y with respect to x, when x = 1 and y = 2, is -1.50 (rounded to 2 decimal places).