Sure, let's find the derivative of the function [tex]\( y = (\csc x + \cot x)^9 \)[/tex].
### Step 1: Identify the outer function
The function given is [tex]\( y = (\csc x + \cot x)^9 \)[/tex]. Here, the outer function is raising [tex]\( (\csc x + \cot x) \)[/tex] to the 9th power. We will apply the chain rule to differentiate this function.
### Step 2: Apply the chain rule
The chain rule states that if you have a composite function [tex]\( y = [u(x)]^n \)[/tex], then:
[tex]\[
\frac{d y}{d x} = n [u(x)]^{n-1} \frac{d u}{d x}
\][/tex]
In this case, [tex]\( u(x) = \csc x + \cot x \)[/tex] and [tex]\( n = 9 \)[/tex].
Let's differentiate [tex]\( u(x) = \csc x + \cot x \)[/tex].
### Step 3: Differentiate [tex]\( u(x) \)[/tex]
To find [tex]\(\frac{d u}{d x}\)[/tex], we need the derivatives of [tex]\(\csc x\)[/tex] and [tex]\(\cot x\)[/tex]:
[tex]\[
\frac{d}{d x}(\csc x) = -\csc x \cot x
\][/tex]
[tex]\[
\frac{d}{d x}(\cot x) = -\csc^2 x
\][/tex]
Therefore,
[tex]\[
\frac{d u}{d x} = \frac{d}{d x}(\csc x + \cot x) = -\csc x \cot x - \csc^2 x
\][/tex]
### Step 4: Combine results using the chain rule
Now, we combine the results from the chain rule:
[tex]\[
\frac{d y}{d x} = 9 (\csc x + \cot x)^8 \cdot \left(-\csc x \cot x - \csc^2 x\right)
\][/tex]
Factor out the negative sign:
[tex]\[
\frac{d y}{d x} = -9 (\csc x + \cot x)^8 \left(\csc x \cot x + \csc^2 x\right)
\][/tex]
### Step 5: Simplify (optional)
Let's simplify the expression inside the parenthesis:
[tex]\[
\csc x \cot x + \csc^2 x = \csc x (\cot x + \csc x)
\][/tex]
Thus, the derivative of the given function is:
[tex]\[
\frac{d y}{d x} = -9 (\csc x + \cot x)^8 \left(\csc x (\cot x + \csc x)\right)
\][/tex]
Or more compactly (substituting the expressions):
[tex]\[
\frac{d y}{d x} = (\cot x + \csc x)^8 (-9 \cot^2 x - 9 \cot x \csc x - 9)
\][/tex]
This is the final answer.
