To differentiate the function [tex]\( y = 7 \csc(x) + 8 \cos(x) \)[/tex], we'll take the derivative of each term separately using standard differentiation rules.
### Step-by-Step Solution
1. Differentiate the first term [tex]\( 7 \csc(x) \)[/tex]:
- The derivative of [tex]\( \csc(x) \)[/tex] is [tex]\( -\csc(x) \cot(x) \)[/tex].
- Applying the constant multiple rule, the derivative of [tex]\( 7 \csc(x) \)[/tex] is:
[tex]\[
7 \cdot \left(-\csc(x) \cot(x)\right) = -7 \csc(x) \cot(x).
\][/tex]
2. Differentiate the second term [tex]\( 8 \cos(x) \)[/tex]:
- The derivative of [tex]\( \cos(x) \)[/tex] is [tex]\( -\sin(x) \)[/tex].
- Applying the constant multiple rule, the derivative of [tex]\( 8 \cos(x) \)[/tex] is:
[tex]\[
8 \cdot \left(-\sin(x)\right) = -8 \sin(x).
\][/tex]
3. Combine the derivatives:
- Adding the derivatives from the two terms, we get:
[tex]\[
-7 \csc(x) \cot(x) - 8 \sin(x).
\][/tex]
Thus, the derivative of the function [tex]\( y = 7 \csc(x) + 8 \cos(x) \)[/tex] is:
[tex]\[
\boxed{-7 \csc(x) \cot(x) - 8 \sin(x)}.
\][/tex]
For convenience, and using trigonometric identities, we can also express this derivative in terms of simpler functions:
[tex]\[
-7 \csc(x) \cot(x) - 8 \sin(x).
\][/tex]
Therefore, the final result is:
[tex]\[
\boxed{-8 \sin(x) - 7 \cot(x) \csc(x)}.
\][/tex]
