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Given
y=3cot(2x), find
(dy)/(dx).
Answer:
(dy)/(dx)=
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Given $y=3 \cot (2 x)$ , find $\frac{d y}{d x}$ . $\newline$ Answer: $\frac{d y}{d x}=$ $\newline$ Submit Answer

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Write a formula for an arithmetic sequence

Full solution

Q. Given

y=3 \cot (2 x)

, find

\frac{d y}{d x}

.

\newline

Answer:

\frac{d y}{d x}=

\newline

Submit Answer

Identify Function and Rules: Identify the function to differentiate and the rules that will be applied. The function is $y = 3\cot(2x)$ , and we will use the chain rule and the derivative of cotangent.
Recall Cotangent Derivative: Recall the derivative of cotangent. The derivative of $\cot(x)$ with respect to $x$ is $-\csc^2(x)$ , where $\csc$ is the cosecant function.
Apply Chain Rule: Apply the chain rule. The chain rule states that the derivative of a composite function $f(g(x))$ is $f'(g(x))g'(x)$ . Here, $f(x) = 3\cot(x)$ and $g(x) = 2x$ . We need to find the derivative of $f$ with respect to $g$ and then multiply by the derivative of $g$ with respect to $x$ .
Differentiate Composite Function: Differentiate $f(g(x)) = 3\cot(g(x))$ with respect to $g$ . The derivative is $-3\csc^2(g(x))$ because the derivative of cotangent is $-\csc^2(x)$ , and we multiply by the constant $3$ .
Differentiate Linear Function: Differentiate $g(x) = 2x$ with respect to $x$ . The derivative is $2$ because the derivative of $x$ with respect to $x$ is $1$ , and we multiply by the constant $2$ .
Combine Derivatives: Combine the derivatives using the chain rule. Multiply the derivative of $f$ with respect to $g$ by the derivative of $g$ with respect to $x$ to get the derivative of $y$ with respect to $x$ . This gives us $\frac{dy}{dx} = -3\csc^2(2x) \times 2$ .
Simplify Final Result: Simplify the expression. Multiply $-3$ by $2$ to get $-6$ . The final derivative is $\frac{dy}{dx} = -6\csc^2(2x)$ .

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