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Given y=-csc(3x), find (dy)/(dx). Answer: (dy)/(dx)=

Given y=−csc⁡(3x) y=-\csc (3 x) , find dydx \frac{d y}{d x} .\newlineAnswer: dydx= \frac{d y}{d x}=

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Q. Given y=−csc⁡(3x) y=-\csc (3 x) , find dydx \frac{d y}{d x} .\newlineAnswer: dydx= \frac{d y}{d x}=
  1. Recall definition of cosecant function: Recall the definition of the cosecant function and its derivative. The cosecant function is defined as csc(x)=1sin⁡(x)\text{csc}(x) = \frac{1}{\sin(x)}, and its derivative with respect to xx is given by ddxcsc(x)=−csc(x)cot(x)\frac{d}{dx}\text{csc}(x) = -\text{csc}(x)\text{cot}(x). We will use this information to find the derivative of y=−csc(3x)y = -\text{csc}(3x) with respect to xx.
  2. Apply chain rule to differentiate: Apply the chain rule to differentiate y=−csc⁡(3x)y = -\csc(3x). The chain rule states that the derivative of a composite function f(g(x))f(g(x)) is f′(g(x))g′(x)f'(g(x))g'(x). In this case, f(x)=−csc⁡(x)f(x) = -\csc(x) and g(x)=3xg(x) = 3x. We need to find the derivative of ff with respect to gg, which is f′(g(x))f'(g(x)), and then multiply it by the derivative of gg with respect to xx, which is f(g(x))f(g(x))00.
  3. Differentiate f(g(x))f(g(x)) with respect to gg: Differentiate f(g(x))=−csc⁡(g(x))f(g(x)) = -\csc(g(x)) with respect to gg. Using the derivative of the cosecant function from Step 11, we get f′(g(x))=−(−csc⁡(g(x))cot⁡(g(x)))=csc⁡(g(x))cot⁡(g(x))f'(g(x)) = -(-\csc(g(x))\cot(g(x))) = \csc(g(x))\cot(g(x)).
  4. Differentiate g(x)g(x) with respect to xx: Differentiate g(x)=3xg(x) = 3x with respect to xx. The derivative of g(x)g(x) with respect to xx is g′(x)=3g'(x) = 3.
  5. Combine results using chain rule: Combine the results from Steps 33 and 44 using the chain rule. Multiply f′(g(x))f'(g(x)) by g′(x)g'(x) to get the derivative of yy with respect to xx. This gives us (dy)/(dx)=csc⁡(3x)cot⁡(3x)×3(dy)/(dx) = \csc(3x)\cot(3x) \times 3.
  6. Simplify the derivative: Simplify the expression for the derivative. Since there is a negative sign in the original function y=−csc⁡(3x)y = -\csc(3x), we need to include this in our final derivative. The correct derivative is dydx=−3csc⁡(3x)cot⁡(3x)\frac{dy}{dx} = -3\csc(3x)\cot(3x).

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