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{:[" 3) "y=tantan^(2)x^(3)],[cot u=x^(3)dv=3x]:}

Find the derivatives of \begin{array} yy=\operatorname{arctan}^{2} x^{3} \\ \cot u=x^{3} d v=3 x\end{array}

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Q. Find the derivatives of \begin{array} yy=\operatorname{arctan}^{2} x^{3} \\ \cot u=x^{3} d v=3 x\end{array}
  1. Identify Functions: Identify the functions and their types.\newliney=tan⁡(tan⁡2(x3))y = \tan(\tan^2(x^3)) is a composite trigonometric function.\newlinecot⁡(u)=x3\cot(u) = x^3 dv=3xdv = 3x is a differential equation involving trigonometric and polynomial components.
  2. Analyze Growth of yy: Analyze the growth of y=tan⁡(tan⁡2(x3))y = \tan(\tan^2(x^3)). As xx increases, x3x^3 increases rapidly. tan⁡2(x3)\tan^2(x^3) oscillates but squares the tangent values, potentially increasing the output range. tan⁡\tan of these values will also oscillate and can grow large.
  3. Analyze Growth of cot⁡(u)\cot(u): Analyze the growth of cot⁡(u)=x3\cot(u) = x^3 dv=3xdv = 3x. This expression is not standard and seems incorrectly formatted. Assuming cot⁡(u)=x3\cot(u) = x^3 and dv=3x dxdv = 3x \, dx, the growth of x3x^3 is cubic, which is a steady increase.

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