To find the derivative of the function [tex]\(\cot(1 + x)\)[/tex] with respect to [tex]\(x\)[/tex], let's go through the steps carefully.
1. Recall the definition of [tex]\(\cot(u)\)[/tex]:
The cotangent function [tex]\(\cot(u)\)[/tex] is defined as the reciprocal of the tangent function: [tex]\(\cot(u) = \frac{1}{\tan(u)}\)[/tex].
2. Use the chain rule for differentiation:
Since [tex]\(\cot(1 + x)\)[/tex] is a composition of functions [tex]\(u = 1 + x\)[/tex] and [tex]\(\cot(u)\)[/tex], we'll apply the chain rule. The derivative of [tex]\(\cot(u)\)[/tex] with respect to [tex]\(u\)[/tex] is [tex]\(-\csc^2(u)\)[/tex].
3. Differentiate with the chain rule:
We need to take the derivative of [tex]\(\cot(1 + x)\)[/tex]:
[tex]\[
\frac{d}{dx} \cot(1 + x) = -\csc^2(1 + x) \cdot \frac{d}{dx}(1 + x)
\][/tex]
4. Differentiate inside the chain:
The derivative of [tex]\(1 + x\)[/tex] with respect to [tex]\(x\)[/tex] is simply [tex]\(1\)[/tex]:
[tex]\[
\frac{d}{dx}(1 + x) = 1
\][/tex]
5. Combine the results from the chain rule:
Putting it all together, we multiply the derivatives:
[tex]\[
\frac{d}{dx} \cot(1 + x) = -\csc^2(1 + x) \cdot 1 = -\csc^2(1 + x)
\][/tex]
6. Express in terms of [tex]\(\cot\)[/tex]:
Recall that [tex]\(\csc^2(u) = 1 + \cot^2(u)\)[/tex]. Applying this identity, we get:
[tex]\[
-\csc^2(1 + x) = - (1 + \cot^2(1 + x))
\][/tex]
Hence, the derivative of [tex]\(\cot(1 + x)\)[/tex] with respect to [tex]\(x\)[/tex] is:
[tex]\[
\boxed{-\cot^2(1 + x) - 1}
\][/tex]
This result matches our calculations and confirms that our steps were followed correctly.
