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Find the derivative of  \cot ^{-1}\left(1+x^2\right) with respect to x^2+x+1

Option: 1

\frac{2 x}{(2 x+1)\left(2+2 x+x^4\right)}


Option: 2

\frac{5 x}{(3 x+1)\left(3+3 x+x^4\right)}


Option: 3

\frac{-2 x}{(2 x+1)\left(2+2 x+x^4\right)}


Option: 4

\frac{-5 x}{(3 x+1)\left(3+3 x+x^4\right)}


Answers (1)

best_answer

Let f(x)=\cot ^{-1}\left(1+x^2\right) and g(x)=x^2+x+1 \begin{aligned} & \frac{\mathrm{d} f}{\mathrm{~d} g}=\frac{f^{\prime}(x)}{g^{\prime}(x)} \\ & f^{\prime}(x)=\frac{-2 x}{2+2 x+x^4} \\ & g^{\prime}(x)=2 x+1 \\ & \frac{f^{\prime}(x)}{g^{\prime}(x)}=\frac{-2 x\left(2+2 x+x^4\right)}{2 x+1}=\frac{-2 x}{(2 x+1)\left(2+2 x+x^4\right)} \end{aligned}

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