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Given 
f(x)=-3cot^(2)(x), find 
f^(')(x).
Answer: 
f^(')(x)=
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Given f(x)=3cot2(x) f(x)=-3 \cot ^{2}(x) , find f(x) f^{\prime}(x) .\newlineAnswer: f(x)= f^{\prime}(x)= \newlineSubmit Answer

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Q. Given f(x)=3cot2(x) f(x)=-3 \cot ^{2}(x) , find f(x) f^{\prime}(x) .\newlineAnswer: f(x)= f^{\prime}(x)= \newlineSubmit Answer
  1. Identify Function & Rule: Identify the function and the rule to differentiate it.\newlineWe have f(x)=3cot2(x)f(x) = -3\cot^2(x), which is a composite function. To differentiate it, we will use the chain rule and the derivative of cot(x)\cot(x), which is csc2(x)-\csc^2(x).
  2. Apply Chain Rule: Apply the chain rule.\newlineThe chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. In this case, the outer function is g(u)=3u2g(u) = -3u^2 and the inner function is u(x)=cot(x)u(x) = \cot(x).
  3. Differentiate Outer Function: Differentiate the outer function g(u)=3u2g(u) = -3u^2 with respect to uu.g(u)=ddu(3u2)=6ug'(u) = \frac{d}{du}(-3u^2) = -6u
  4. Differentiate Inner Function: Differentiate the inner function u(x)=cot(x)u(x) = \cot(x) with respect to xx.u(x)=ddx(cot(x))=csc2(x)u'(x) = \frac{d}{dx}(\cot(x)) = -\csc^2(x)
  5. Apply Chain Rule Again: Apply the chain rule using the derivatives from steps 33 and 44.\newlinef(x)=g(u(x))u(x)=(6cot(x))(csc2(x))f'(x) = g'(u(x)) \cdot u'(x) = (-6 \cdot \cot(x)) \cdot (-\csc^2(x))
  6. Simplify Expression: Simplify the expression. f(x)=6cot(x)csc2(x)f'(x) = 6 \cdot \cot(x) \cdot \csc^2(x)

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