13-26 Implicit differentiation Carry out the following steps.
b. Find the slope of the curve at the given point.x⁴+y⁴ = 2;(1,−1)

Verified step by step guidance

1. Start by differentiating both sides of the equation with respect to x. The equation is x⁴ + y⁴ = 2. Use implicit differentiation, which involves differentiating y with respect to x as well.
2. Differentiate x⁴ with respect to x, which gives 4x³. For y⁴, apply the chain rule: differentiate y⁴ with respect to y to get 4y³, then multiply by dy/dx (the derivative of y with respect to x).
3. Set up the equation from the differentiation: 4x³ + 4y³(dy/dx) = 0. This equation represents the derivative of the original equation.
4. Solve for dy/dx, which represents the slope of the curve. Rearrange the equation to isolate dy/dx: dy/dx = -4x³ / 4y³.
5. Substitute the given point (1, -1) into the equation for dy/dx. Replace x with 1 and y with -1 to find the slope at this specific point.

Key Concepts

Implicit Differentiation

Implicit differentiation is a technique used to differentiate equations where the dependent and independent variables are not explicitly separated. Instead of solving for one variable in terms of the other, we differentiate both sides of the equation with respect to the independent variable, applying the chain rule as necessary. This method is particularly useful for curves defined by equations like x⁴ + y⁴ = 2, where y cannot be easily isolated.

Slope of a Curve

The slope of a curve at a given point represents the rate of change of the y-coordinate with respect to the x-coordinate at that point. Mathematically, it is found by evaluating the derivative of the function at the specified point. In the context of implicit differentiation, once we find dy/dx, we can substitute the coordinates of the point (1, -1) to determine the slope of the curve at that location.

Chain Rule

The chain rule is a fundamental principle in calculus used to differentiate composite functions. It states that if a function y is defined as a function of u, which in turn is a function of x, then the derivative of y with respect to x can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to x. This rule is essential in implicit differentiation, as it allows us to differentiate terms involving y when applying the derivative to both sides of an equation.