Differentiating Implicitly


Use implicit differentiation to find dy/dx in Exercises 1–14.


x³ + y³ = 18xy

Verified step by step guidance

1. Start by differentiating both sides of the equation with respect to x. The equation is x³ + y³ = 18xy.
2. Differentiate x³ with respect to x, which gives 3x².
3. Differentiate y³ with respect to x using implicit differentiation. Since y is a function of x, this gives 3y²(dy/dx).
4. Differentiate 18xy with respect to x. Use the product rule: differentiate 18x to get 18, and multiply by y, then differentiate y to get dy/dx and multiply by 18x. This results in 18y + 18x(dy/dx).
5. Set the derivatives equal: 3x² + 3y²(dy/dx) = 18y + 18x(dy/dx). Solve for dy/dx by isolating terms involving dy/dx on one side of the equation.

Key Concepts

Implicit Differentiation

Implicit differentiation is a technique used to differentiate equations where the dependent variable is not isolated on one side. Instead of solving for y explicitly, we differentiate both sides of the equation with respect to x, applying the chain rule to terms involving y. This allows us to find dy/dx without needing to express y as a function of x.

Chain Rule

The chain rule is a fundamental principle in calculus that allows us to differentiate composite functions. When differentiating a function of another function, the chain rule states that the derivative is the product of the derivative of the outer function and the derivative of the inner function. In implicit differentiation, this is particularly important when differentiating terms involving y, as we treat dy/dx as the derivative of y with respect to x.

Solving for dy/dx

After applying implicit differentiation to an equation, the next step is to isolate dy/dx. This often involves rearranging the differentiated equation to express dy/dx in terms of x and y. Once isolated, dy/dx provides the slope of the tangent line to the curve defined by the original equation at any point (x, y).