Exercise 3

\[g(t)=\sin(t^2) - \cos(t^2)\] \[g^\prime (t)=\cos(t^2)\, 2t + sin (t^2)\, 2t = 2t\,(\cos(t^2) + \sin (t^2) )\]

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\[\Theta(\alpha)=\mathrm{e}^{3\alpha}+3\alpha \ln (3\alpha)\] \[\Theta^\prime (\alpha)=\mathrm{e}^{3\alpha} \, 3 + 3 ( \ln (3\alpha) + \alpha \frac{1}{3\alpha} \, 3)\] \[ = 3 \mathrm{e}^{3\alpha} + 3 ( \ln (3\alpha) + 1)\] \[= 3 ( \mathrm{e}^{3\alpha} + \ln (3\alpha) + 1)\]

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\[x(t)=\sin^2(t^2)-\cos^2(t^2)\] \[x^\prime (t) = 2 \sin(t^2) \cos(t^2) \, 2t - (2 \cos(t^2) (-\sin(t^2))\, 2t )\] \[= 8t \sin(t^2) \cos(t^2)\]