27–40. Implicit differentiation Use implicit differentiation to find dy/dx.
6x³+7y³ = 13xy

Verified step by step guidance

Start by differentiating both sides of the equation with respect to x. The equation is 6x³ + 7y³ = 13xy.

Differentiate the left side: The derivative of 6x³ with respect to x is 18x². For 7y³, use the chain rule: the derivative is 21y²(dy/dx).

Differentiate the right side: Use the product rule for 13xy. The derivative is 13(dy/dx)x + 13y.

Set up the equation from the derivatives: 18x² + 21y²(dy/dx) = 13(dy/dx)x + 13y.

Solve for dy/dx: Rearrange the equation to isolate dy/dx on one side. This involves factoring out dy/dx and simplifying the expression.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 equations that are difficult or impossible to rearrange.

Chain Rule

The chain rule is a fundamental principle in calculus that allows us 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 is essential in implicit differentiation, where we often encounter terms involving both x and y.

Derivative Notation (dy/dx)

The notation dy/dx represents the derivative of y with respect to x, indicating the rate of change of y as x changes. In the context of implicit differentiation, finding dy/dx involves isolating the derivative of y in terms of x and potentially other constants. Understanding this notation is crucial for interpreting the results of differentiation and applying them to problems involving rates of change.

27–40. Implicit differentiation Use implicit differentiation to find dy/dx.6x³+7y³ = 13xy