13-26 Implicit differentiation Carry out the following steps.
a. Use implicit differentiation to find dy/dx.
x⁴+y⁴ = 2;(1,−1)

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Start by differentiating both sides of the equation \(x^4 + y^4 = 2\) with respect to \(x\). Remember that \(y\) is a function of \(x\), so you'll need to use implicit differentiation for terms involving \(y\).

Differentiate \(x^4\) with respect to \(x\) to get \(4x^3\).

Differentiate \(y^4\) with respect to \(x\) using the chain rule. This gives \(4y^3 \frac{dy}{dx}\).

Set the derivative of the left side equal to the derivative of the right side, which is zero, since the derivative of a constant is zero: \(4x^3 + 4y^3 \frac{dy}{dx} = 0\).

Solve for \(\frac{dy}{dx}\) by isolating it on one side of the equation: \(\frac{dy}{dx} = -\frac{x^3}{y^3}\). Finally, substitute the point \((1, -1)\) into the expression to find the specific value of \(\frac{dy}{dx}\) at that point.

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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 y in terms of x, we differentiate both sides of the equation with respect to x, treating y as a function of x. This method allows us to find the derivative dy/dx without isolating y, which is particularly useful for complex equations.

Chain Rule

The chain rule is a fundamental principle in calculus that allows us to differentiate composite functions. When using implicit differentiation, we apply the chain rule to account for the derivative of y with respect to x, denoted as dy/dx. This means that when differentiating terms involving y, we multiply by dy/dx to correctly represent the relationship between the variables.

Evaluating Derivatives at a Point

After finding the expression for dy/dx through implicit differentiation, we often need to evaluate this derivative at a specific point, such as (1, -1) in this case. This involves substituting the x and y values into the derived expression to find the slope of the tangent line at that point. This step is crucial for understanding the behavior of the function at specific coordinates.

13-26 Implicit differentiation Carry out the following steps.a. Use implicit differentiation to find dy/dx.x⁴+y⁴ = 2;(1,−1)