51–56. Second derivatives Find d²y/dx².
x⁴+y⁴ = 64

Verified step by step guidance

1. Start by differentiating the given equation implicitly with respect to x. The equation is x⁴ + y⁴ = 64.
2. Differentiate both sides of the equation: d/dx(x⁴ + y⁴) = d/dx(64).
3. Apply the power rule to differentiate x⁴, which gives 4x³. For y⁴, use implicit differentiation: 4y³(dy/dx).
4. Set the derivative of the constant 64 to zero, as the derivative of a constant is zero.
5. Now, solve for dy/dx: 4x³ + 4y³(dy/dx) = 0. Rearrange to find dy/dx = -x³/y³.

Key Concepts

Implicit Differentiation

Implicit differentiation is a technique used to differentiate equations where the dependent and independent variables are not explicitly separated. In this case, the equation x⁴ + y⁴ = 64 involves both x and y, requiring us to differentiate with respect to x while treating y as a function of x. This method allows us to find the first derivative dy/dx, which is essential for further calculations.

First Derivative

The first derivative, denoted as dy/dx, represents the rate of change of the dependent variable y with respect to the independent variable x. It provides information about the slope of the tangent line to the curve defined by the equation. In the context of finding the second derivative, the first derivative is a crucial step, as it will be differentiated again to obtain d²y/dx².

Second Derivative

The second derivative, denoted as d²y/dx², measures the rate of change of the first derivative. It provides insights into the curvature of the function, indicating whether the function is concave up or down. In this problem, calculating the second derivative involves differentiating the first derivative obtained from implicit differentiation, which helps analyze the behavior of the function around specific points.