Given x^5+5xy-y^2=y+7, use implicit differentiation to find dy/dx.
dy/dx=?

We are given the equation:

\[
x^5 + 5xy - y^2 = y + 7
\]

To find \(\frac{dy}{dx}\), we'll use **implicit differentiation**, treating \(y\) as a function of \(x\) (i.e., \(y = y(x)\)).

### Step 1: Differentiate both sides with respect to \(x\)

#### Left-hand side:

- The derivative of \(x^5\) with respect to \(x\) is \(5x^4\).
- The derivative of \(5xy\) involves the product rule:
  \[
  \frac{d}{dx}(5xy) = 5 \left( y + x \frac{dy}{dx} \right) = 5y + 5x \frac{dy}{dx}
  \]
- The derivative of \(-y^2\) using the chain rule:
  \[
  \frac{d}{dx}(-y^2) = -2y \frac{dy}{dx}
  \]

So, differentiating the left-hand side, we get:
\[
5x^4 + 5y + 5x \frac{dy}{dx} - 2y \frac{dy}{dx}
\]

#### Right-hand side:

- The derivative of \(y\) is \(\frac{dy}{dx}\).
- The derivative of the constant \(7\) is \(0\).

So, differentiating the right-hand side, we get:
\[
\frac{dy}{dx}
\]

### Step 2: Set up the equation

Now, equate the derivatives from both sides:
\[
5x^4 + 5y + 5x \frac{dy}{dx} - 2y \frac{dy}{dx} = \frac{dy}{dx}
\]

### Step 3: Solve for \(\frac{dy}{dx}\)

First, group all the terms with \(\frac{dy}{dx}\) on one side:
\[
5x^4 + 5y = \frac{dy}{dx} - 5x \frac{dy}{dx} + 2y \frac{dy}{dx}
\]

Factor out \(\frac{dy}{dx}\) on the right-hand side:
\[
5x^4 + 5y = \frac{dy}{dx} \left( 1 - 5x + 2y \right)
\]

Finally, solve for \(\frac{dy}{dx}\):
\[
\frac{dy}{dx} = \frac{5x^4 + 5y}{1 - 5x + 2y}
\]

Thus, the derivative \(\frac{dy}{dx}\) is:

\[
\boxed{\frac{5x^4 + 5y}{1 - 5x + 2y}}
\]

Would you like further details or have any questions?

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Here are 5 related questions for you to explore:
1. How does implicit differentiation differ from explicit differentiation?
2. Can you find \(\frac{d^2y}{dx^2}\) for the same equation?
3. What would change if the equation involved a trigonometric function like \(\sin(y)\)?
4. How does the chain rule help in differentiating composite functions?
5. What is the significance of the product rule in calculus?

**Tip:** When differentiating implicitly, always treat \(y\) as a function of \(x\), and apply the chain rule when differentiating terms involving \(y\).

Given x^5 + 5xy - y^2 = y + 7, use implicit differentiation to find dy/dx.

Learn how to use implicit differentiation to find dy/dx for the equation x^5 + 5xy - y^2 = y + 7. This detailed step-by-step solution involves key calculus concepts such as the product rule and chain rule, ideal for Calculus I students.

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