To perform implicit differentiation on the equation:

\[
x^8 y^9 - y = x
\]

we will differentiate both sides with respect to \(x\), treating \(y\) as a function of \(x\) (i.e., \(y = y(x)\)).

### Step-by-Step Solution:

1. **Differentiate the left-hand side:**

\[
\frac{d}{dx}\left(x^8 y^9 - y\right)
\]

- For the term \(x^8 y^9\), we apply the product rule:
\[
\frac{d}{dx}(x^8 y^9) = \frac{d}{dx}(x^8) \cdot y^9 + x^8 \cdot \frac{d}{dx}(y^9)
\]
\[
= 8x^7 y^9 + x^8 \cdot 9y^8 \frac{dy}{dx}
\]

- For the term \(-y\), we differentiate with respect to \(x\), treating \(y\) as a function of \(x\):
\[
\frac{d}{dx}(-y) = -\frac{dy}{dx}
\]

Thus, the derivative of the left-hand side is:
\[
8x^7 y^9 + 9x^8 y^8 \frac{dy}{dx} - \frac{dy}{dx}
\]

2. **Differentiate the right-hand side:**

\[
\frac{d}{dx}(x) = 1
\]

Now, we have the equation:
\[
8x^7 y^9 + 9x^8 y^8 \frac{dy}{dx} - \frac{dy}{dx} = 1
\]

3. **Solve for \(\frac{dy}{dx}\):**

Group all the terms involving \(\frac{dy}{dx}\):
\[
9x^8 y^8 \frac{dy}{dx} - \frac{dy}{dx} = 1 - 8x^7 y^9
\]

Factor out \(\frac{dy}{dx}\) on the left-hand side:
\[
\frac{dy}{dx}(9x^8 y^8 - 1) = 1 - 8x^7 y^9
\]

Finally, solve for \(\frac{dy}{dx}\):
\[
\frac{dy}{dx} = \frac{1 - 8x^7 y^9}{9x^8 y^8 - 1}
\]

Thus, the derivative is:
\[
\boxed{\frac{dy}{dx} = \frac{1 - 8x^7 y^9}{9x^8 y^8 - 1}}
\]

Would you like further details or have any questions about the steps?

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Here are 5 related questions to expand your understanding:

1. How would you apply implicit differentiation to \(x^3y^2 + \sin(y) = \ln(x)\)?
2. What is the significance of using the product rule when implicitly differentiating products of \(x\) and \(y\)?
3. Can you find the second derivative of \(y\) with respect to \(x\) for the original equation?
4. What happens to the solution if \(x = 1\) and \(y = 1\) in the given equation?
5. How does implicit differentiation differ from explicit differentiation?

**Tip**: When implicitly differentiating, always remember to apply the chain rule to the terms involving \(y\), since \(y\) is treated as a function of \(x\).

implicit differentiation of: x^8y^9 - y = x

Learn how to apply implicit differentiation to the equation x^8y^9 - y = x. This problem involves the product rule and chain rule, providing a detailed step-by-step solution. Suitable for advanced high school or early college students studying calculus.

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