https://file.aiquickdraw.com/mathbot/file/172895955710301344eqo.jpg

To differentiate the given function, let's break it down.

You need to find the derivative of the function:

\[
f(x) = \frac{4(x^2 + 3) \cot(x)}{2 - \cos(x) \csc(x)}
\]

This is a quotient, so we'll apply the quotient rule. The quotient rule states:

\[
\frac{d}{dx} \left(\frac{u(x)}{v(x)}\right) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}
\]

### Step 1: Identify \(u(x)\) and \(v(x)\)
Let:
- \( u(x) = 4(x^2 + 3) \cot(x) \)
- \( v(x) = 2 - \cos(x) \csc(x) \)

### Step 2: Differentiate \(u(x)\) and \(v(x)\)

1. **Derivative of \( u(x) = 4(x^2 + 3) \cot(x) \)**

We use the product rule here:
\[
u'(x) = 4 \left[ \frac{d}{dx}(x^2 + 3) \cdot \cot(x) + (x^2 + 3) \cdot \frac{d}{dx} \cot(x) \right]
\]
- \(\frac{d}{dx} (x^2 + 3) = 2x\)
- \(\frac{d}{dx} \cot(x) = -\csc^2(x)\)

Thus,
\[
u'(x) = 4 \left[ 2x \cot(x) - (x^2 + 3) \csc^2(x) \right]
\]

2. **Derivative of \( v(x) = 2 - \cos(x) \csc(x) \)**

We apply the product rule:
\[
v'(x) = \frac{d}{dx} \left(2 - \cos(x) \csc(x)\right)
\]
- \(\frac{d}{dx} 2 = 0\)
- \(\frac{d}{dx} \left(\cos(x) \csc(x)\right)\) requires the product rule:
\[
\frac{d}{dx} \left( \cos(x) \csc(x) \right) = \frac{d}{dx} \cos(x) \cdot \csc(x) + \cos(x) \cdot \frac{d}{dx} \csc(x)
\]
- \(\frac{d}{dx} \cos(x) = -\sin(x)\)
- \(\frac{d}{dx} \csc(x) = -\csc(x) \cot(x)\)

Thus:
\[
v'(x) = 0 - \left[-\sin(x) \csc(x) + \cos(x) (-\csc(x) \cot(x))\right]
\]
Simplifying:
\[
v'(x) = \sin(x) - \cos(x) \csc(x) \cot(x)
\]

### Step 3: Apply the Quotient Rule

Now apply the quotient rule:

\[
f'(x) = \frac{u'(x) v(x) - u(x) v'(x)}{[v(x)]^2}
\]

Substitute the expressions for \(u'(x)\), \(u(x)\), \(v'(x)\), and \(v(x)\) into this formula, and simplify accordingly.

Would you like me to compute the entire simplification step-by-step, or would you like to review this first?

---

Here are 5 related questions you can explore:
1. What is the derivative of \(\cot(x)\) and \(\csc(x)\)?
2. How does the product rule apply when differentiating two functions?
3. Can you solve similar problems involving trigonometric functions?
4. What are some common simplifications for trigonometric identities?
5. How does the quotient rule differ from the product rule?

**Tip:** Always simplify your trigonometric expressions first before applying differentiation rules for easier calculation.

Differentiate f(x) = (4(x^2 + 3) * cot(x)) / (2 - cos(x) * csc(x)).

Learn how to differentiate the function f(x) = (4(x^2 + 3) * cot(x)) / (2 - cos(x) * csc(x)) using the quotient rule and product rule. This solution walks through the step-by-step process, including trigonometric identities and derivative rules. Suitable for advanced high school or college-level calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation