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Chapter 2: Problem 56

Differentiate implicitly to find \(d^{2} y / d x^{2}\). $$x^{2}-y^{2}=5$$

Short Answer

Expert verified
\( \frac{d^2 y}{d x^2} = \frac{y^2 - x^2}{y^3} \)

Step by step solution

01

Differentiate both sides with respect to x

Start by differentiating the given equation implicitly with respect to x. The original equation is:\[ x^2 - y^2 = 5 \]Differentiating both sides with respect to x gives:\[ \frac{d}{dx}(x^2) - \frac{d}{dx}(y^2) = \frac{d}{dx}(5) \]This simplifies to:\[ 2x - 2y \frac{dy}{dx} = 0 \]
02

Solve for \frac{dy}{dx}

From the equation, isolate the term involving \frac{dy}{dx}:\[ 2x - 2y \frac{dy}{dx} = 0 \]Add \2y \frac{dy}{dx} to both sides:\[ 2x = 2y \frac{dy}{dx} \]Divide by 2y to get:\[ \frac{dy}{dx} = \frac{x}{y} \]
03

Differentiate again with respect to x

Differentiate \( \frac{dy}{dx} = \frac{x}{y} \) implicitly again with respect to x to find \( \frac{d^2 y}{d x^2} \):\[ \frac{d}{dx} \bigg( \frac{dy}{dx} \bigg) = \frac{d}{dx} \bigg( \frac{x}{y} \bigg) \]Use the quotient rule on the right side:\[ \frac{d}{dx} \bigg( \frac{x}{y} \bigg) = \frac{y \frac{d}{dx}(x) - x \frac{d}{dx}(y)}{y^2} \]Since \( \frac{d}{dx}(x) = 1 \) and \( \frac{d}{dx}(y) = \frac{dy}{dx} \), substitute these values:\[ \frac{d}{dx} \bigg( \frac{dy}{dx} \bigg) = \frac{y \times 1 - x \times \frac{dy}{dx}}{y^2} \]Replacing \( \frac{dy}{dx} \) with \( \frac{x}{y} \):\[ = \frac{y - x \times \frac{x}{y}}{y^2} \]Simplify the expression:\[ = \frac{y^2 - x^2}{y^3} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

second derivative
Finding the second derivative involves differentiating the first derivative a second time. In this case, we start with the given equation: :x^2 - y^2 = 5:. Differentiating it once, we get the first derivative, :dy/dx.: The goal is to differentiate a second time to find :d^2y/dx^2:. This second derivative tells us how the rate of change of the function's slope (first derivative) changes with respect to x. In essence, it gives us insights into the concavity and behavior of the original function.
quotient rule
The Quotient Rule is used to differentiate functions that are a ratio of two other functions. When we have a function in the form of :u/v: where both u and v are functions of x, the rule states that the derivative is given by:: (u/v)' = (v*u' - u*v') / v^2:.After finding : dy/dx = x/y:: we need to differentiate :x/y: using the quotient rule to find :d^2y/dx^2:. Remembering the quotient rule will help you compute these kinds of derivatives efficiently.
implicit differentiation steps
Implicit differentiation is used when you have a relationship between x and y that is not explicitly solved for either variable. Here are the steps:
1. Differentiate both sides of the equation with respect to x, applying the chain rule where necessary.
2. Collect all :dy/dx: terms on one side of the equation.
3. Solve for :dy/dx:. For the second derivative, we take the first derivative we found, :x/y:, and differentiate it again using the quotient rule. Step-by-step application of implicit differentiation ensures we accurately determine how x and y change relative to each other.

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