find y'' if x^4+y^4=16

To find the second derivative of y with respect to x, we'll need to differentiate the given equation twice. Let's go through the step-by-step process.

Step 1: Differentiate the equation once with respect to x.
To do this, we need to apply the chain rule. The chain rule states that if we have a composite function, such as y = f(g(x)), then the derivative dy/dx can be found by multiplying the derivative of the outer function f'(g(x)) with the derivative of the inner function g'(x).

Differentiating x⁴ + y⁴ = 16 with respect to x:
4x³ + 4y³ * dy/dx = 0 (since the derivative of a constant is zero)

Step 2: Solve for dy/dx.
To solve for dy/dx, we isolate it in the equation obtained from Step 1.

4y³ * dy/dx = -4x³
dy/dx = -4x³ / 4y³
Simplifying further, we get:
dy/dx = -x³ / y³

Step 3: Differentiate dy/dx with respect to x.
To find the second derivative, we differentiate dy/dx obtained from Step 2 using the quotient rule. The quotient rule states that if we have a function in the form of f(x) / g(x), then the derivative d²y/dx² can be found as follows:

d²y/dx² = (g(x) * d²f(x)/dx² - f(x) * d²g(x)/dx²) / (g(x))²

Let's differentiate dy/dx = -x³ / y³ with respect to x:

Let f(x) = -x³ and g(x) = y³

d²f(x)/dx² = -6x
d²g(x)/dx² = 6y * dy/dx (applying the chain rule)

Substituting these into the quotient rule formula, we have:

d²y/dx² = (y³ * -6x - (-x³) * (6y * dy/dx)) / (y³)²
= (-6xy³ + 6x³y * dy/dx) / y⁶

Since we know dy/dx from Step 2, we can substitute it into the equation:

d²y/dx² = (-6xy³ + 6x³y * (-x³ / y³)) / y⁶
= (-6xy³ - 6x⁴) / y⁵

Therefore, the second derivative of y with respect to x is (-6xy³ - 6x⁴) / y⁵.