https://file.aiquickdraw.com/mathbot/file/172791901046313dmljs8.jpg

To show that:

\[
\frac{d}{dx} (\cot x) = -\csc^2 x
\]

### Step-by-step differentiation:
1. **Express \(\cot x\)**:
   \[
   \cot x = \frac{\cos x}{\sin x}
   \]
   Now we need to differentiate this using the **quotient rule**.

2. **Quotient Rule**: 
   If \( f(x) = \frac{g(x)}{h(x)} \), then:
   \[
   \frac{d}{dx} \left( \frac{g(x)}{h(x)} \right) = \frac{h(x) g'(x) - g(x) h'(x)}{[h(x)]^2}
   \]
   Here, \( g(x) = \cos x \) and \( h(x) = \sin x \).

3. **Differentiate \(\cos x\) and \(\sin x\)**:
   \[
   \frac{d}{dx} (\cos x) = -\sin x, \quad \frac{d}{dx} (\sin x) = \cos x
   \]

4. **Apply the quotient rule**:
   \[
   \frac{d}{dx} (\cot x) = \frac{\sin x \cdot (-\sin x) - \cos x \cdot \cos x}{(\sin x)^2}
   \]
   Simplifying the numerator:
   \[
   = \frac{-\sin^2 x - \cos^2 x}{\sin^2 x}
   \]

5. **Use the Pythagorean identity**: 
   \[
   \sin^2 x + \cos^2 x = 1
   \]
   Therefore, the expression becomes:
   \[
   = \frac{-1}{\sin^2 x}
   \]

6. **Simplify**:
   \[
   \frac{-1}{\sin^2 x} = -\csc^2 x
   \]

Thus, we have shown that:

\[
\frac{d}{dx} (\cot x) = -\csc^2 x
\]

Would you like more details on any part of this solution?

### Related questions:
1. How do you differentiate other trigonometric functions like \(\sec x\) or \(\csc x\)?
2. What is the derivative of \(\tan x\), and how does it compare to \(\cot x\)?
3. Can you apply the quotient rule to non-trigonometric functions?
4. How does the chain rule apply to trigonometric functions?
5. What are the integration counterparts of these trigonometric derivatives?

### Tip:
Memorizing the derivatives of basic trigonometric functions will save you a lot of time in calculus problems.

Show that the derivative of cot(x) is -csc^2(x).

Learn how to differentiate cot(x) step-by-step using the quotient rule and Pythagorean identity. The solution shows that the derivative of cot(x) is -csc^2(x), covering key trigonometric and calculus concepts.

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