Show that y^''=(-a c(m-1) x^m-2)/(b^2 y^2 m-1) if a x^m+b y^m=c

Name: Problem 81, Chapter 3: Applied Calculus For Business, Economics, and Finance
Uploaded: 2021-08-13T18:46:10-08:00
Duration: 1 min 16 s
Channel: Carson Merrill
Description: Show that y^''=(-a c(m-1) x^m-2)/(b^2 y^2 m-1) if a x^m+b y^m=c

step 1 The following problem, we want to show that Y double prime equals a negative AC times m minus 1

Show that y^''=(-a c(m-1) x^m-2)/(b^2 y^2 m-1) if a...

Key Concepts

Implicit Differentiation

This is a method used when a function is defined implicitly rather than explicitly. Instead of expressing y directly in terms of x, both sides of the equation are differentiated with respect to x, treating y as a function of x. This allows one to obtain the derivatives even when the equation cannot be easily solved for y.

Chain Rule

The chain rule is essential when differentiating composite functions, especially when differentiating terms involving y, where y itself is a function of x. In the context of an implicit equation, the chain rule helps in appropriately handling the derivative of functions that depend on y, ensuring that all implicit dependencies on x are accounted for.

Differentiation of Power Functions

When differentiating expressions like ax^m or by^m, the power rule is applied, which states that the derivative of x^m is m*x^(m-1). This concept is crucial for handling each term in an equation where variables are raised to a power, allowing for the correct computation of derivatives in the differentiation process.

Higher-Order Derivatives

After finding the first derivative using implicit differentiation, the process is repeated to find the second derivative (y''). This involves differentiating the first derivative while keeping in mind both the product/quotient rules and the chain rule. Higher-order derivatives provide insights into the concavity and the behavior of the original function.

Algebraic Manipulation

Once differentiation is performed, algebraic manipulation is used to simplify the resulting expressions. This includes factoring, combining like terms, and manipulating exponents to achieve a neatly simplified form for the derivative. Such skills are vital for transforming a complex derivative result into a more interpretable and elegant expression.

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