Question

Find $y^{\prime \prime}$ at (2,1) if $2 x^{2} y-4 y^{3}=4$ (see Problem 37 ).

   Find $y^{\prime \prime}$ at (2,1) if $2 x^{2} y-4 y^{3}=4$ (see Problem 37 ).

Calculus

Dale Varberg, Edwin… 9th Edition

Chapter 2, Problem 39

Instant Answer

Solved by Expert Carson Merrill

08/03/2021

Step 1

The chain rule is used for the term $2x^2y$ and the product rule is used for the term $-4y^3$. The derivative of a constant is zero. So, we have: \[2x^2y' + 4xy - 12y^2y' = 0\] Show more…

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Find $y^{\prime \prime}$ at (2,1) if $2 x^{2} y-4 y^{3}=4$ (see Problem 37 ).

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Key Concepts

Second Derivative in Implicit Differentiation

After finding the first derivative using implicit differentiation, obtaining the second derivative involves differentiating the first derivative. This process typically requires the chain rule and product rule again and may result in an expression that contains both the independent and dependent variables. Evaluating the second derivative at a specific point involves substituting the values of both variables.

Chain Rule

The chain rule is a fundamental tool for differentiating composite functions. In implicit differentiation, when a dependent variable is involved in a composite function, the chain rule is used to account for the inner derivative of the dependent variable with respect to the independent variable.

Implicit Differentiation

This is a technique used to differentiate equations where the dependent variable is not explicitly solved in terms of the independent variable. Instead of rearranging the equation into a standard form, both sides of the equation are differentiated with respect to the independent variable, treating the dependent variable as an implicit function of the independent variable.

Product Rule

The product rule is a method used when differentiating products of two functions. In the context of implicit differentiation, it is applied when the product of functions involving both the independent and dependent variables is present, ensuring that every part of the product is properly differentiated.

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