To find \(\frac{dy}{dx}\) from the given equation \( x \cos(y) + y \cos(x) = 1 \), we will implicitly differentiate both sides of the equation with respect to \(x\).

### Step 1: Differentiate the left side
The given equation is:

\[
x \cos(y) + y \cos(x) = 1
\]

We'll apply the product rule and chain rule where necessary.

1. Differentiate \(x \cos(y)\) with respect to \(x\):
   \[
   \frac{d}{dx} (x \cos(y)) = \cos(y) + x (-\sin(y)) \frac{dy}{dx}
   \]

2. Differentiate \(y \cos(x)\) with respect to \(x\):
   \[
   \frac{d}{dx} (y \cos(x)) = \frac{dy}{dx} \cos(x) + y (-\sin(x))
   \]

### Step 2: Differentiate the right side
The right-hand side is just a constant \(1\), so its derivative is 0:
\[
\frac{d}{dx}(1) = 0
\]

### Step 3: Put it all together
Now, combining all the derivatives:
\[
\cos(y) + x (-\sin(y)) \frac{dy}{dx} + \frac{dy}{dx} \cos(x) + y (-\sin(x)) = 0
\]

### Step 4: Solve for \(\frac{dy}{dx}\)
Now we collect all the terms involving \(\frac{dy}{dx}\) on one side:
\[
x (-\sin(y)) \frac{dy}{dx} + \frac{dy}{dx} \cos(x) = -\cos(y) + y \sin(x)
\]

Factor out \(\frac{dy}{dx}\):
\[
\frac{dy}{dx} \left( x (-\sin(y)) + \cos(x) \right) = -\cos(y) + y \sin(x)
\]

Finally, solve for \(\frac{dy}{dx}\):
\[
\frac{dy}{dx} = \frac{-\cos(y) + y \sin(x)}{x (-\sin(y)) + \cos(x)}
\]

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

\[
\frac{dy}{dx} = \frac{-\cos(y) + y \sin(x)}{x (-\sin(y)) + \cos(x)}
\]

---

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

Here are some related questions you could explore:
1. What is implicit differentiation, and when is it used?
2. How does the product rule apply in differentiation?
3. What is the chain rule, and how does it work with functions of multiple variables?
4. Can you solve for \(y(x)\) explicitly given the original equation?
5. How does the derivative \(\frac{dy}{dx}\) behave for specific values of \(x\) and \(y\)?

**Tip:** Implicit differentiation is particularly useful when you have an equation relating \(x\) and \(y\) but cannot easily solve for one variable in terms of the other.

Explore the detailed solution for finding dy/dx from the implicit equation x cos(y) + y cos(x) = 1. This problem involves key calculus concepts like implicit differentiation, product rule, and chain rule. Suitable for high school or early college calculus students.

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