Certainly! Let's differentiate the function [tex]\( f(x) = -\cot^2(x) \)[/tex] step-by-step.
1. Write down the function:
[tex]\[
f(x) = -\cot^2(x)
\][/tex]
2. Recall the chain rule:
The chain rule states that if you have a composite function [tex]\( g(h(x)) \)[/tex], then the derivative [tex]\( g(h(x)) \)[/tex] with respect to [tex]\( x \)[/tex] is given by:
[tex]\[
\frac{d}{dx} g(h(x)) = g'(h(x)) \cdot h'(x)
\][/tex]
3. Identify the outer and inner functions:
In this case, our function is [tex]\( f(x) = -(\cot(x))^2 \)[/tex]. Here, the outer function is [tex]\( -u^2 \)[/tex] where [tex]\( u = \cot(x) \)[/tex].
4. Differentiate the outer function:
First, differentiate the outer function [tex]\( -u^2 \)[/tex] with respect to [tex]\( u \)[/tex]:
[tex]\[
\frac{d}{du}(-u^2) = -2u
\][/tex]
5. Differentiate the inner function:
Now, differentiate the inner function [tex]\( \cot(x) \)[/tex] with respect to [tex]\( x \)[/tex]:
[tex]\[
\frac{d}{dx}(\cot(x)) = -\csc^2(x)
\][/tex]
6. Combine using the chain rule:
According to the chain rule, the derivative of [tex]\( f(x) = -\cot^2(x) \)[/tex] with respect to [tex]\( x \)[/tex] is:
[tex]\[
f'(x) = \frac{d}{dx}[-(\cot(x))^2] = -2\cot(x) \cdot \left(-\csc^2(x)\right)
\][/tex]
7. Simplify the expression:
Combine the terms and simplify:
[tex]\[
f'(x) = -2\cot(x) \cdot (-\csc^2(x)) = 2\cot(x) \csc^2(x)
\][/tex]
However, considering the result from the original provided computational answer, let's ensure consistency:
[tex]\[
f'(x) = -2\cot(x) \left(\cot^2(x) + 1\right)
\][/tex]
Thus, the differentiation of [tex]\( f(x) = -\cot^2(x) \)[/tex] yields:
[tex]\[
f'(x) = -(-2\cot^2(x) - 2) \cot(x)
\][/tex]
Therefore, our final simplified result for the derivative is:
[tex]\[
f'(x) = -(-2\cot^2(x) - 2) \cot(x)
\][/tex]
This is the derivative of [tex]\(-\cot^2(x)\)[/tex].
