Find the second derivative when the equation is 5xy + y^2 =5

To find the second derivative \(\frac{d^2y}{dx^2}\) when the equation is \(5xy + y^2 = 5\), we follow these steps:

### Step 1: Differentiate the equation implicitly with respect to \(x\)

Given:
\[
5xy + y^2 = 5
\]

First, differentiate both sides with respect to \(x\):

- For the term \(5xy\), apply the product rule: 
  \[
  \frac{d}{dx}(5xy) = 5\left(x \frac{dy}{dx} + y\right) = 5\left(\frac{dy}{dx} \cdot x + y\right)
  \]
- For the term \(y^2\), apply the chain rule: 
  \[
  \frac{d}{dx}(y^2) = 2y \frac{dy}{dx}
  \]
- The right-hand side, being a constant, differentiates to 0.

So, the derivative of the entire equation is:
\[
5\left(x \frac{dy}{dx} + y\right) + 2y \frac{dy}{dx} = 0
\]

### Step 2: Solve for \(\frac{dy}{dx}\)

Rearrange the equation to solve for \(\frac{dy}{dx}\):
\[
5x \frac{dy}{dx} + 5y + 2y \frac{dy}{dx} = 0
\]
\[
\frac{dy}{dx}(5x + 2y) = -5y
\]
\[
\frac{dy}{dx} = \frac{-5y}{5x + 2y}
\]

### Step 3: Differentiate again to find \(\frac{d^2y}{dx^2}\)

Differentiate \(\frac{dy}{dx}\) implicitly with respect to \(x\):

\[
\frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{-5y}{5x + 2y}\right)
\]

This requires the quotient rule:
\[
\frac{d^2y}{dx^2} = \frac{(5x + 2y) \frac{d}{dx}(-5y) - (-5y) \frac{d}{dx}(5x + 2y)}{(5x + 2y)^2}
\]

Calculate \(\frac{d}{dx}(-5y)\) and \(\frac{d}{dx}(5x + 2y)\):
- \(\frac{d}{dx}(-5y) = -5\frac{dy}{dx}\)
- \(\frac{d}{dx}(5x + 2y) = 5 + 2\frac{dy}{dx}\)

Substitute into the equation:
\[
\frac{d^2y}{dx^2} = \frac{(5x + 2y)(-5\frac{dy}{dx}) - (-5y)(5 + 2\frac{dy}{dx})}{(5x + 2y)^2}
\]

Substitute \(\frac{dy}{dx} = \frac{-5y}{5x + 2y}\) into the equation:

\[
\frac{d^2y}{dx^2} = \frac{-5(5x + 2y)\left(\frac{-5y}{5x + 2y}\right) - (-5y)\left(5 + 2\left(\frac{-5y}{5x + 2y}\right)\right)}{(5x + 2y)^2}
\]

Simplify each term:

- First term: \(-5(5x + 2y)\left(\frac{-5y}{5x + 2y}\right) = 25y\)
- Second term: 
  \[
  -(-5y)\left(5 - \frac{10y}{5x + 2y}\right) = 25y - \frac{50y^2}{5x + 2y}
  \]

Combine the terms:

\[
\frac{d^2y}{dx^2} = \frac{25y + 25y - \frac{50y^2}{5x + 2y}}{(5x + 2y)^2}
\]
\[
\frac{d^2y}{dx^2} = \frac{50y - \frac{50y^2}{5x + 2y}}{(5x + 2y)^2}
\]

This can be further simplified, but that’s the expression for the second derivative.

Would you like any part of this explanation to be expanded, or do you have any questions?

### Related Questions:
1. What is the significance of the second derivative in terms of the curvature of the graph?
2. How does the quotient rule apply in the context of implicit differentiation?
3. What are some common errors when applying the chain rule in implicit differentiation?
4. How would the solution change if we modified the equation to include a different power of \(y\)?
5. Can implicit differentiation be applied to equations with more complex forms, such as those involving logarithms?
6. How does implicit differentiation relate to related rates problems?
7. Could you show an example of solving a second derivative for a parametric equation?
8. What is the geometric interpretation of the first and second derivatives?

### Tip:
When working with implicit differentiation, it's often useful to double-check your application of product and quotient rules, as these are common areas where mistakes can occur.

Learn how to find the second derivative \(\frac{d^2y}{dx^2}\) for the equation 5xy + y^2 = 5 using implicit differentiation and the quotient rule. This problem explores advanced calculus concepts and techniques. Suitable for college-level mathematics students.

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