Exercise 1

\[ f(t)=t^4+3t^3-12t^2+t-6 \] \[ f^{\prime}(t) = 4t^3+9t^2-24t+1 \]

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\[g(x)=\sin x - \cos x\] \[g^\prime(x) = \cos x + \sin x \]

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\[\Theta(\alpha)=\mathrm{e}^\alpha +\alpha \ln \alpha\] For the expression \(\alpha \ln \alpha\) we use the product rule: \[\Theta^\prime(\alpha)=\mathrm{e}^\alpha + \ln \alpha\ + \alpha \frac{1}{\alpha} = \mathrm{e}^\alpha + \ln \alpha\ + 1 \]

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\[x(t)=\sin^2 t - \cos t\] For \(\sin^2 t\) we need the chain rule. Let \(f(t) := \sin t\): \[ x(t) = f(t)^2 - \cos t \] \[x^\prime(t) = 2\, f(t)\, f^\prime(t) + \sin t = 2 \sin t \cos t + \sin t\]