Question

Given the equation x² + 4y² = 16, what is the expression that represents the second derivative with respect to x, or d²/dx², of the function defined by this equation?"

Answer 1

The second derivative of the function defined implicitly by the equation x² + 4y² = 16 involves implicit differentiation, the Chain Rule, and the Quotient Rule to obtain an expression for d²y/dx².

Step-by-step explanation:

Finding the Second Derivative

To find the second derivative of the function implicitly defined by the equation x² + 4y² = 16, we must first implicitly differentiate with respect to x to get the first derivative dy/dx and then differentiate again to obtain the second derivative d²y/dx².

Starting with the given equation:

x² + 4y² = 16

First differentiation with respect to x (using the Chain Rule for y since y is a function of x):

1. 2x + 8y · (dy/dx) = 0.
2. Solve for dy/dx: (dy/dx) = -2x/(8y) = -x/(4y).

Then, differentiate dy/dx with respect to x to get d²y/dx²:

1. (d/dx) (-x/(4y)) = (d/dx) (-1/4 · x/y).
2. Using the Quotient Rule, we have:
3. d²y/dx² = (-1/4 · (y · (d/dx)(x) - x · (d/dx)(y))) / y².
4. After substituting (dy/dx) = -x/(4y), simplify to find the second derivative.

The expression for the second derivative involves applying the differentiation rules dealing with implicit differentiation, using the Chain Rule and the Quotient Rule.