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To find \(\frac{dy}{dx}\) using implicit differentiation for the equation:

\[
\cos y + 9 = x
\]

we proceed as follows:

1. Differentiate both sides of the equation with respect to \(x\). Remember, \(y\) is a function of \(x\), so we'll apply the chain rule when differentiating \(\cos y\).

\[
\frac{d}{dx}(\cos y + 9) = \frac{d}{dx}(x)
\]

2. Differentiating the left side term-by-term:
   - The derivative of \(\cos y\) with respect to \(x\) is \(-\sin y \cdot \frac{dy}{dx}\).
   - The derivative of \(9\) (a constant) is \(0\).
   - The derivative of \(x\) with respect to \(x\) is \(1\).

So we get:
\[
-\sin y \cdot \frac{dy}{dx} = 1
\]

3. Now, solve for \(\frac{dy}{dx}\) by isolating it on one side:

\[
\frac{dy}{dx} = -\frac{1}{\sin y}
\]

### Final Answer:
\[
\frac{dy}{dx} = -\frac{1}{\sin y}
\]

Would you like more details on any step, or do you have any questions?

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Here are some related questions that expand on this topic:

1. What is implicit differentiation, and when is it used?
2. How would \(\frac{dy}{dx}\) change if the equation was \(\sin y + 9 = x\)?
3. What is the geometric interpretation of finding \(\frac{dy}{dx}\) in implicit differentiation?
4. How can implicit differentiation be applied to equations involving higher-order derivatives?
5. How does implicit differentiation differ from explicit differentiation?

**Tip:** Always apply the chain rule when differentiating terms with \(y\) in terms of \(x\) since \(y\) is implicitly a function of \(x\).

Calculate dy/dx using implicit differentiation for the equation: cos(y) + 9 = x

Learn how to find dy/dx using implicit differentiation for the equation cos(y) + 9 = x. This problem explores trigonometric differentiation and the chain rule, essential for advanced calculus students.

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