Evaluate differentiation of y with respect to x:
`(2x+1)^(-3)`

Explanation

To evaluate the differentiation of \( y \) with respect to \( x \) for the function \( y = (2x + 1)^{-3} \), we will use the chain rule. Here’s a step-by-step solution:

### Step 1: Identify the function
We start with the function:
\[ y = (2x + 1)^{-3} \]

### Step 2: Apply the chain rule
According to the chain rule, if \( y = u^n \), where \( u = 2x + 1 \) and \( n = -3 \), the derivative \( \frac{dy}{dx} \) is given by:
\[ \frac{dy}{dx} = n \cdot u^{n-1} \cdot \frac{du}{dx} \]
In our case, \( n = -3 \) and \( u = 2x + 1 \).

### Step 3: Differentiate \( u \)
Now we need to find \( \frac{du}{dx} \):
\[ u = 2x + 1 \implies \frac{du}{dx} = 2 \]

### Step 4: Substitute into the derivative formula
Now we can substitute \( n \), \( u \), and \( \frac{du}{dx} \) into the formula:
\[ \frac{dy}{dx} = -3 \cdot (2x + 1)^{-3 - 1} \cdot 2 \]
This simplifies to:
\[ \frac{dy}{dx} = -3 \cdot (2x + 1)^{-4} \cdot 2 \]

### Step 5: Simplify the expression
Now we can simplify:
\[ \frac{dy}{dx} = -6 \cdot (2x + 1)^{-4} \]

### Final Result
Thus, the derivative of \( y \) with respect to \( x \) is:
\[ \frac{dy}{dx} = -\frac{6}{(2x + 1)^{4}} \]

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