To find the derivative of the given function \( y = \frac{5(x^{1/2})(\ln x)}{\tan x} \), we will apply both the **quotient rule** and the **product rule**.

### Step-by-Step Solution:

1. **Rewrite the function:**

   The function can be written as:
   \[
   y = \frac{5 \sqrt{x} \ln x}{\tan x}
   \]
   where \( \sqrt{x} = x^{1/2} \).

2. **Apply the quotient rule:**
   The quotient rule is given by:
   \[
   \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}
   \]
   Here:
   - \( u = 5 \sqrt{x} \ln x \)
   - \( v = \tan x \)

3. **Differentiate the numerator (u):**
   Apply the product rule to \( u = 5 \sqrt{x} \ln x \).

   The product rule states:
   \[
   (fg)' = f'g + fg'
   \]
   Let:
   - \( f = 5 \sqrt{x} = 5x^{1/2} \)
   - \( g = \ln x \)

   Now, find the derivatives of \( f \) and \( g \):
   - \( f' = 5 \cdot \frac{1}{2} x^{-1/2} = \frac{5}{2 \sqrt{x}} \)
   - \( g' = \frac{1}{x} \)

   Using the product rule:
   \[
   u' = \left(\frac{5}{2 \sqrt{x}}\right) \ln x + 5 \sqrt{x} \cdot \frac{1}{x}
   \]
   Simplify the second term:
   \[
   u' = \frac{5 \ln x}{2 \sqrt{x}} + \frac{5}{\sqrt{x}}
   \]

4. **Differentiate the denominator (v):**
   \[
   v = \tan x \quad \text{so} \quad v' = \sec^2 x
   \]

5. **Apply the quotient rule:**
   Now use the quotient rule formula:
   \[
   y' = \frac{u'v - uv'}{v^2}
   \]
   Substituting the values:
   \[
   y' = \frac{\left( \frac{5 \ln x}{2 \sqrt{x}} + \frac{5}{\sqrt{x}} \right) \tan x - 5 \sqrt{x} \ln x \cdot \sec^2 x}{\tan^2 x}
   \]

6. **Simplify the result:**
   This is the derivative of the given function. It can be simplified further, but the key steps have been completed.

Let me know if you would like me to continue simplifying, provide more details, or if you have any questions.

### Related Questions:
1. How do we differentiate a product of functions?
2. What are the key steps to applying the quotient rule?
3. Can you explain how to differentiate logarithmic functions?
4. How does the derivative of trigonometric functions like \( \tan x \) work?
5. Why does the chain rule play a role in differentiating functions like \( \tan x \)?

### Tip:
When applying the quotient rule, always differentiate the numerator and denominator separately before combining them, and simplify carefully to avoid mistakes.

Learn how to find the derivative of the function y = [5(sqrt(x)lnx)]/(tanx) using the quotient rule and product rule. This problem involves logarithmic differentiation and calculus concepts, ideal for students in Grades 11-12 or those studying AP Calculus.

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