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Prove that if y=csc x thenv y’=-cscxcotx

You have uploaded an image showing the function \( f(x) = \frac{x^5 - 20x^2 + 10x}{4x} \).

It seems you are also asking for a proof of the derivative of \( y = \csc(x) \), which is:

\[
y' = -\csc(x) \cot(x)
\]

Let's go over the proof for this derivative step by step.

### Step-by-Step Proof:

1. **Definition of \( \csc(x) \)**:
   \[
   \csc(x) = \frac{1}{\sin(x)}
   \]
   So, we want to differentiate \( y = \frac{1}{\sin(x)} \).

2. **Apply the Quotient Rule**:
   The quotient rule states that if \( y = \frac{u(x)}{v(x)} \), then:
   \[
   y' = \frac{v(x) u'(x) - u(x) v'(x)}{v(x)^2}
   \]
   Here, \( u(x) = 1 \) and \( v(x) = \sin(x) \). So, let's differentiate:

   - \( u'(x) = 0 \) (since \( u(x) = 1 \) is constant)
   - \( v'(x) = \cos(x) \) (the derivative of \( \sin(x) \))

3. **Apply to the Function**:
   \[
   y' = \frac{\sin(x) \cdot 0 - 1 \cdot \cos(x)}{\sin^2(x)}
   \]
   Simplifying:
   \[
   y' = \frac{-\cos(x)}{\sin^2(x)}
   \]

4. **Rewrite Using Trig Identities**:
   - Recall that \( \frac{1}{\sin(x)} = \csc(x) \) and \( \frac{\cos(x)}{\sin(x)} = \cot(x) \).
   So:
   \[
   y' = -\csc(x) \cdot \cot(x)
   \]

Thus, we have proved that if \( y = \csc(x) \), then \( y' = -\csc(x) \cot(x) \).

---

Would you like any more details on this proof? Feel free to ask for clarification.

### Related Questions:
1. What is the derivative of \( \cot(x) \)?
2. Can you derive the derivative of \( \sec(x) \)?
3. How do you apply the chain rule when differentiating trigonometric functions?
4. What are the key differences between the product rule and quotient rule?
5. How do you integrate \( \csc(x) \)?

### Tip:
Memorizing the derivatives of common trigonometric functions like \( \sin(x) \), \( \cos(x) \), and their reciprocals can greatly speed up solving calculus problems!

Prove the derivative of the function f(x) = (x^5 - 20x^2 + 10x) / (4x)

This problem involves finding the derivative of the rational function f(x) = (x^5 - 20x^2 + 10x) / (4x). Explore key concepts in calculus such as the quotient rule and power rule, with a detailed, step-by-step explanation. Suitable for high school calculus students.

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