Sabendo que \(\vec{F}(u)=\left\langle u^3sqrt{u}\right\rangle \mathrm{m}(\mathrm{u})=\sqrt{u}\), assinale a alternativa que apresenta a derivada da...

Primeiro, vamos encontrar a função \(\vec{G}(u)\) substituindo \(\vec{F}(u)\) em sua expressão: \begin{align*} \vec{G}(u) &= 32 \vec{F}(m(u)) \\ &= 32 \left\langle (m(u))^3\sqrt{m(u)}\right\rangle \mathrm{m}((m(u))) \\ &= 32 \left\langle (u^2)^3\sqrt{u^2}\right\rangle \mathrm{m}(u^2) \\ &= 32 \left\langle u^6|u|\right\rangle \mathrm{m}(u^2) \\ &= 32 u^7 \left\langle |u|\right\rangle \mathrm{m}(u^2) \\ \end{align*} Agora, podemos encontrar a derivada de \(\vec{G}(u)\) no ponto \(u=4\) aplicando a regra da cadeia e a regra do produto: \begin{align*} \frac{d\vec{G}}{du}(u) &= \frac{d}{du}\left[32 u^7 \left\langle |u|\right\rangle \mathrm{m}(u^2)\right] \\ &= 32 \frac{d}{du}\left[u^7 \left\langle |u|\right\rangle \mathrm{m}(u^2)\right] \\ &= 32 \left[\frac{d}{du}(u^7) \left\langle |u|\right\rangle \mathrm{m}(u^2) + u^7 \frac{d}{du}\left(\left\langle |u|\right\rangle \mathrm{m}(u^2)\right)\right] \\ &= 32 \left[7u^6 \left\langle |u|\right\rangle \mathrm{m}(u^2) + u^7 \frac{d}{du}\left(\left\langle |u|\right\rangle \mathrm{m}(u^2)\right)\right] \\ &= 32 \left[7u^6 \left\langle |u|\right\rangle \mathrm{m}(u^2) + u^7 \left(\frac{d}{du}\left\langle |u|\right\rangle \mathrm{m}(u^2) + \left\langle |u|\right\rangle \frac{d}{du}\mathrm{m}(u^2)\right)\right] \\ &= 32 \left[7u^6 \left\langle |u|\right\rangle \mathrm{m}(u^2) + u^7 \left(\frac{u}{|u|}\mathrm{m}(u^2) + \left\langle |u|\right\rangle \frac{d}{du}\mathrm{m}(u^2)\right)\right] \\ \end{align*} Agora, podemos substituir \(u=4\) na expressão acima para encontrar a derivada de \(\vec{G}(u)\) no ponto \(u=4\): \begin{align*} \frac{d\vec{G}}{du}(4) &= 32 \left[7(4)^6 \left\langle |4|\right\rangle \mathrm{m}(4^2) + (4)^7 \left(\frac{4}{|4|}\mathrm{m}(4^2) + \left\langle |4|\right\rangle \frac{d}{du}\mathrm{m}(4^2)\right)\right] \\ &= 32 \left[7(4)^6 \cdot 4 \cdot \mathrm{m}(16) + (4)^7 \left(\frac{4}{4}\mathrm{m}(16) + 1 \cdot \frac{d}{du}\mathrm{m}(16)\right)\right] \\ &= 32 \left[7(4)^6 \cdot 4 \cdot \mathrm{m}(16) + (4)^7 \cdot 2 \cdot \frac{d}{du}\mathrm{m}(16)\right] \\ &= 32 \left[7(4)^6 \cdot 4 \cdot \mathrm{m}(16) + (4)^7 \cdot 2 \cdot \frac{d}{du}\left(\frac{1}{2}(1+\mathrm{sgn}(u))\right)\right] \\ &= 32 \left[7(4)^6 \cdot 4 \cdot \mathrm{m}(16) + (4)^7 \cdot 2 \cdot \frac{d}{du}\left(\frac{1}{2}(1+\mathrm{sgn}(4))\right)\right] \\ &= 32 \left[7(4)^6 \cdot 4 \cdot \mathrm{m}(16) + (4)^7 \cdot 2 \cdot \frac{d}{du}\left(\frac{1}{2}(1+1)\right)\right] \\ &= 32 \left[7(4)^6 \cdot 4 \cdot \mathrm{m}(16) + (4)^7 \cdot 2 \cdot 0\right] \\ &= 32 \cdot 7(4)^6 \cdot 4 \cdot \mathrm{m}(16) \\ &= 28672 \mathrm{m}(16) \\ \end{align*} Portanto, a alternativa correta é: \textbf{D) \(28672 \mathrm{m}(16)\)}