13-26 Implicit differentiation Carry out the following steps.
b. Find the slope of the curve at the given point.
(x+y)^2/3=y; (4, 4)

Verified step by step guidance

1. Start by differentiating both sides of the equation \((x + y)^{2/3} = y\) with respect to \(x\). Remember that \(y\) is a function of \(x\), so you'll need to use implicit differentiation.
2. Apply the chain rule to differentiate the left side: \(\frac{d}{dx}((x + y)^{2/3}) = \frac{2}{3}(x + y)^{-1/3} \cdot (1 + \frac{dy}{dx})\).
3. Differentiate the right side with respect to \(x\): \(\frac{d}{dx}(y) = \frac{dy}{dx}\).
4. Set the derivatives equal to each other: \(\frac{2}{3}(x + y)^{-1/3} \cdot (1 + \frac{dy}{dx}) = \frac{dy}{dx}\).
5. Solve for \(\frac{dy}{dx}\) by isolating it on one side of the equation. Substitute \(x = 4\) and \(y = 4\) into the equation to find the slope of the curve at the point \((4, 4)\).

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 that cannot be easily rearranged.

Slope of a Curve

The slope of a curve at a given point represents the rate of change of the dependent variable with respect to the independent variable at that point. Mathematically, it is found by evaluating the derivative of the function at the specified coordinates. In the context of implicit differentiation, the slope can be determined by substituting the coordinates of the point into the derivative obtained from the implicit differentiation process.

Chain Rule

The chain rule is a fundamental principle in calculus used to differentiate composite functions. It states that if a function is composed of two or more functions, the derivative of the outer function is multiplied by the derivative of the inner function. In implicit differentiation, the chain rule is applied when differentiating terms involving the dependent variable, ensuring that the derivative accounts for the relationship between the variables.