Question

the due date has passed, but late work is permitted un
7.
let
$f(x) = 5\csc(9x)$

$f(x) = \boxed{}$
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Explanation:

Step1: Recall derivative of csc(u)

The derivative of $\csc(u)$ with respect to $x$ is $-\csc(u)\cot(u)\cdot u'$, where $u$ is a function of $x$. Here, $f(x) = 5\csc(9x)$, so we use the constant multiple rule and the chain rule. Let $u = 9x$, then $u' = 9$.

Step2: Apply constant multiple and chain rule

First, the constant multiple rule: the derivative of $5\csc(9x)$ is $5$ times the derivative of $\csc(9x)$. Using the chain rule, the derivative of $\csc(9x)$ is $-\csc(9x)\cot(9x)\cdot 9$. So multiplying by $5$, we get $5\times(-9\csc(9x)\cot(9x))$.

Step3: Simplify the expression

Simplifying $5\times(-9\csc(9x)\cot(9x))$ gives $-45\csc(9x)\cot(9x)$.

Answer:

$-45\csc(9x)\cot(9x)$