Q. Given f(x)=−3cot2(x), find f′(x).Answer: f′(x)=Submit Answer
Identify Function & Rule: Identify the function and the rule to differentiate it.We have f(x)=−3cot2(x), which is a composite function. To differentiate it, we will use the chain rule and the derivative of cot(x), which is −csc2(x).
Apply Chain Rule: Apply the chain rule.The 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)=−3u2 and the inner function is u(x)=cot(x).
Differentiate Outer Function: Differentiate the outer function g(u)=−3u2 with respect to u.g′(u)=dud(−3u2)=−6u
Differentiate Inner Function: Differentiate the inner function u(x)=cot(x) with respect to x.u′(x)=dxd(cot(x))=−csc2(x)
Apply Chain Rule Again: Apply the chain rule using the derivatives from steps 3 and 4.f′(x)=g′(u(x))⋅u′(x)=(−6⋅cot(x))⋅(−csc2(x))
Simplify Expression: Simplify the expression. f′(x)=6⋅cot(x)⋅csc2(x)
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