Given f(x)=3csc(3x), find f^(')(x). Answer: f^(')(x)=

Given Function: We are given the function f(x) = 3csc(3x) and we need to find its derivative f'(x). The csc(x) function is the reciprocal of the sin(x) function, so csc(x) = 1/sin(x). The derivative of csc(u) with respect to u is -csc(u)cot(u), where cot(u) is the reciprocal of tan(u), or cot(u) = cos(u)/sin(u). We will use the chain rule to differentiate f(x) = 3csc(3x), where the outer function is 3csc(u) and the inner function is u = 3x.Derivative of Outer Function: First, let's find the derivative of the outer function with respect to u, which is 3csc(u). The derivative of csc(u) is -csc(u)cot(u), so the derivative of 3csc(u) is -3csc(u)cot(u).Derivative of Inner Function: Next, we need to find the derivative of the inner function u = 3x with respect to x. The derivative of 3x with respect to x is 3.Apply Chain Rule: Now, we apply the chain rule, which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. Therefore, f'(x) = -3csc(3x)cot(3x) * 3.Simplify Expression: Simplify the expression for f'(x) by multiplying the constants together. f'(x) = -9csc(3x)cot(3x).

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Given
f(x)=3csc(3x), find
f^(')(x).
Answer:
f^(')(x)=

Given $f(x)=3 \csc (3 x)$ , find $f^{\prime}(x)$ . $\newline$ Answer: $f^{\prime}(x)=$

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Precalculus

Find trigonometric ratios using multiple identities

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Q. Given

f(x)=3 \csc (3 x)

, find

f^{\prime}(x)

\newline

Answer:

f^{\prime}(x)=

Given Function: We are given the function $f(x) = 3\csc(3x)$ and we need to find its derivative $f'(x)$ . The $\csc(x)$ function is the reciprocal of the $\sin(x)$ function, so $\csc(x) = \frac{1}{\sin(x)}$ . The derivative of $\csc(u)$ with respect to $u$ is $-\csc(u)\cot(u)$ , where $\cot(u)$ is the reciprocal of $\tan(u)$ , or $f'(x)$ $0$ . We will use the chain rule to differentiate $f(x) = 3\csc(3x)$ , where the outer function is $f'(x)$ $2$ and the inner function is $f'(x)$ $3$ .
Derivative of Outer Function: First, let's find the derivative of the outer function with respect to $u$ , which is $3\csc(u)$ . The derivative of $\csc(u)$ is $-\csc(u)\cot(u)$ , so the derivative of $3\csc(u)$ is $-3\csc(u)\cot(u)$ .
Derivative of Inner Function: Next, we need to find the derivative of the inner function $u = 3x$ with respect to $x$ . The derivative of $3x$ with respect to $x$ is $3$ .
Apply Chain Rule: Now, we apply the chain rule, which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. Therefore, $f'(x) = -3\csc(3x)\cot(3x) \times 3$ .
Simplify Expression: Simplify the expression for $f'(x)$ by multiplying the constants together. $f'(x) = -9\csc(3x)\cot(3x)$ .