Differentiate. f(x) = sin(x) + 2/5 cot(x)

To differentiate the function \( f(x) = \sin(x) + \frac{2}{5} \cot(x) \), let's apply the rules of differentiation step by step:

### Step 1: Differentiate \( \sin(x) \)
The derivative of \( \sin(x) \) is:
\[
\frac{d}{dx}[\sin(x)] = \cos(x)
\]

### Step 2: Differentiate \( \frac{2}{5} \cot(x) \)
The derivative of \( \cot(x) \) is:
\[
\frac{d}{dx}[\cot(x)] = -\csc^2(x)
\]
Thus, the derivative of \( \frac{2}{5} \cot(x) \) is:
\[
\frac{2}{5} \cdot \frac{d}{dx}[\cot(x)] = \frac{2}{5} \cdot (-\csc^2(x)) = -\frac{2}{5} \csc^2(x)
\]

### Final Answer:
Now, combine the results:
\[
f'(x) = \cos(x) - \frac{2}{5} \csc^2(x)
\]

Let me know if you'd like further clarification or details.

### Related Questions:
1. How would you differentiate \( f(x) = \sin(x) \cdot \cot(x) \)?
2. What is the derivative of \( \sin^2(x) + \cot^2(x) \)?
3. How do you differentiate \( \cos(x) + \csc(x) \)?
4. What is the second derivative of \( f(x) = \sin(x) + \frac{2}{5} \cot(x) \)?
5. How do you differentiate \( \tan(x) + \frac{1}{3} \sec(x) \)?

### Tip:
For trigonometric functions, it helps to remember the basic derivatives like \( \frac{d}{dx}[\tan(x)] = \sec^2(x) \) and \( \frac{d}{dx}[\cot(x)] = -\csc^2(x) \).

Learn how to differentiate the function f(x) = sin(x) + (2/5) cot(x). This problem explores the use of trigonometric differentiation, applying rules to calculate derivatives of sine and cotangent functions. Suitable for students in Grades 11-12.

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