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To find the second derivative \( y'' \) for the function given:

\[
y = \frac{1}{4} \cot(2x + 1),
\]

we will first find the first derivative \( y' \), and then differentiate it again to obtain \( y'' \).

### Step 1: Find the first derivative \( y' \).

Given:
\[
y = \frac{1}{4} \cot(2x + 1),
\]
we need to use the chain rule and the derivative of the cotangent function. The derivative of \( \cot(u) \) with respect to \( u \) is \( -\csc^2(u) \), and then we need to multiply by the derivative of \( u \) with respect to \( x \).

Let \( u = 2x + 1 \), then:
\[
\frac{dy}{dx} = \frac{1}{4} \left( -\csc^2(2x + 1) \right) \cdot \frac{d}{dx}(2x + 1).
\]
The derivative of \( 2x + 1 \) is 2, so:
\[
y' = \frac{1}{4} \cdot (-\csc^2(2x + 1)) \cdot 2 = -\frac{1}{2} \csc^2(2x + 1).
\]

### Step 2: Find the second derivative \( y'' \).

Now, we need to differentiate \( y' = -\frac{1}{2} \csc^2(2x + 1) \) with respect to \( x \).

We use the chain rule again, where the derivative of \( \csc^2(u) \) is \( -2 \csc^2(u) \cot(u) \), and we need to multiply by the derivative of \( u = 2x + 1 \), which is 2.

So, differentiating \( y' \), we get:
\[
y'' = -\frac{1}{2} \cdot (-2 \csc^2(2x + 1) \cot(2x + 1)) \cdot 2.
\]

Simplifying, we find:
\[
y'' = 2 \csc^2(2x + 1) \cot(2x + 1).
\]

Thus, the second derivative is:
\[
y'' = 2 \csc^2(2x + 1) \cot(2x + 1).
\]

Would you like further details on these steps, or do you have any questions?

Here are five related questions to expand on this topic:
1. How do you find higher-order derivatives of trigonometric functions?
2. What are the derivatives of other trigonometric functions like \( \sin \), \( \cos \), or \( \tan \)?
3. How does the chain rule apply to finding derivatives of composite functions?
4. What is the significance of the second derivative in analyzing a function's graph?
5. Can you apply these differentiation techniques to functions involving exponential or logarithmic expressions?

**Tip:** Remember that for functions involving trigonometric expressions, the chain rule is crucial when the argument of the trigonometric function is not simply \( x \).

Find the second derivative of y = 1/4 cot(2x + 1) by applying the chain rule and trigonometric derivative formulas. Learn step-by-step how to differentiate composite functions involving cotangent. This is a typical advanced calculus problem suitable for Grades 11-12.

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