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We are tasked with differentiating the given function:

\[
y = 5 \cos x \cot x
\]

### Step-by-step differentiation:

We will use the product rule since the expression is a product of two functions, \( \cos x \) and \( \cot x \).

The product rule states:

\[
\frac{d}{dx}[uv] = u'v + uv'
\]

Here, \( u = \cos x \) and \( v = \cot x \).

1. Differentiate \( u = \cos x \):
   \[
   u' = -\sin x
   \]

2. Differentiate \( v = \cot x \):
   \[
   v' = -\csc^2 x
   \]

Now, apply the product rule:

\[
\frac{dy}{dx} = 5 \left( (-\sin x)(\cot x) + (\cos x)(-\csc^2 x) \right)
\]

Simplifying:

\[
\frac{dy}{dx} = 5 \left( -\sin x \cot x - \cos x \csc^2 x \right)
\]

Thus, the derivative is:

\[
\boxed{\frac{dy}{dx} = 5 \left( -\sin x \cot x - \cos x \csc^2 x \right)}
\]

Would you like further clarification on any step?

### Follow-up questions:
1. What is the product rule in differentiation?
2. How do you differentiate trigonometric functions like \( \cot x \)?
3. What is the role of the chain rule in differentiation?
4. Can you use the quotient rule to differentiate \( \cot x \)?
5. How would this problem change if the function involved other trigonometric terms?

**Tip:** The product rule is particularly useful when two functions of \( x \) are multiplied together. Always break it into two parts.

Find the derivative of y = 5 cos(x) cot(x) with respect to x.

Learn how to differentiate the trigonometric function y = 5 cos(x) cot(x) using the product rule. This detailed solution involves applying derivatives of cos(x) and cot(x) and simplifying the result.

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