Derive a seguinte função g(x) = 20.Sen²(x/2)
OBS.: Explique detalhadanente cada passo.
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![\\ \displaystyle \mathsf{g(x) = 20 \cdot \sin^2 \left ( \frac{x}{2} \right )} \\\\\\ \mathsf{g(x) = 20 \cdot \left [ \sin \left ( \frac{x}{2} \right ) \right ]^2} \\\\\\ \mathsf{g'(x) = 20 \cdot 2 \cdot \left [ \sin \left ( \frac{x}{2} \right ) \right ]^{2 - 1} \cdot \left [ \sin \left ( \frac{x}{2} \right ) \right ]'} \\\\\\ \mathsf{g'(x) = 20 \cdot 2 \cdot \left [ \sin \left ( \frac{x}{2} \right ) \right ]^1 \cdot \left [ \cos \left ( \frac{x}{2} \right ) \cdot \left ( \frac{x}{2} \right )' \right ]} \\\\\\ \mathsf{g'(x) = 20 \cdot 2 \cdot \sin \left ( \frac{x}{2} \right ) \cdot \left [ \cos \left ( \frac{x}{2} \right ) \cdot \left ( \frac{1}{2} \right ) \right ]} \\\\\\ \mathsf{g'(x) = 20 \cdot 2 \cdot \sin \left ( \frac{x}{2} \right ) \cdot \cos \left ( \frac{x}{2} \right ) \cdot \frac{1}{2}} \\\\\\ \mathsf{g'(x) = \frac{20 \cdot 2}{2} \cdot \sin \left ( \frac{x}{2} \right ) \cdot \cos \left ( \frac{x}{2} \right )} \\\\\\ \boxed{\mathsf{g'(x) = 20 \cdot \sin \left ( \frac{x}{2} \right ) \cdot \cos \left ( \frac{x}{2} \right )}}](https://tex.z-dn.net/?f=%5C%5C+%5Cdisplaystyle+%5Cmathsf%7Bg%28x%29+%3D+20+%5Ccdot+%5Csin%5E2+%5Cleft+%28+%5Cfrac%7Bx%7D%7B2%7D+%5Cright+%29%7D+%5C%5C%5C%5C%5C%5C+%5Cmathsf%7Bg%28x%29+%3D+20+%5Ccdot+%5Cleft+%5B+%5Csin+%5Cleft+%28+%5Cfrac%7Bx%7D%7B2%7D+%5Cright+%29+%5Cright+%5D%5E2%7D+%5C%5C%5C%5C%5C%5C+%5Cmathsf%7Bg%27%28x%29+%3D+20+%5Ccdot+2+%5Ccdot+%5Cleft+%5B+%5Csin+%5Cleft+%28+%5Cfrac%7Bx%7D%7B2%7D+%5Cright+%29+%5Cright+%5D%5E%7B2+-+1%7D+%5Ccdot+%5Cleft+%5B+%5Csin+%5Cleft+%28+%5Cfrac%7Bx%7D%7B2%7D+%5Cright+%29+%5Cright+%5D%27%7D+%5C%5C%5C%5C%5C%5C+%5Cmathsf%7Bg%27%28x%29+%3D+20+%5Ccdot+2+%5Ccdot+%5Cleft+%5B+%5Csin+%5Cleft+%28+%5Cfrac%7Bx%7D%7B2%7D+%5Cright+%29+%5Cright+%5D%5E1+%5Ccdot+%5Cleft+%5B+%5Ccos+%5Cleft+%28+%5Cfrac%7Bx%7D%7B2%7D+%5Cright+%29+%5Ccdot+%5Cleft+%28+%5Cfrac%7Bx%7D%7B2%7D+%5Cright+%29%27+%5Cright+%5D%7D+%5C%5C%5C%5C%5C%5C+%5Cmathsf%7Bg%27%28x%29+%3D+20+%5Ccdot+2+%5Ccdot+%5Csin+%5Cleft+%28+%5Cfrac%7Bx%7D%7B2%7D+%5Cright+%29+%5Ccdot+%5Cleft+%5B+%5Ccos+%5Cleft+%28+%5Cfrac%7Bx%7D%7B2%7D+%5Cright+%29+%5Ccdot+%5Cleft+%28+%5Cfrac%7B1%7D%7B2%7D+%5Cright+%29+%5Cright+%5D%7D+%5C%5C%5C%5C%5C%5C+%5Cmathsf%7Bg%27%28x%29+%3D+20+%5Ccdot+2+%5Ccdot+%5Csin+%5Cleft+%28+%5Cfrac%7Bx%7D%7B2%7D+%5Cright+%29+%5Ccdot+%5Ccos+%5Cleft+%28+%5Cfrac%7Bx%7D%7B2%7D+%5Cright+%29+%5Ccdot+%5Cfrac%7B1%7D%7B2%7D%7D+%5C%5C%5C%5C%5C%5C+%5Cmathsf%7Bg%27%28x%29+%3D+%5Cfrac%7B20+%5Ccdot+2%7D%7B2%7D+%5Ccdot+%5Csin+%5Cleft+%28+%5Cfrac%7Bx%7D%7B2%7D+%5Cright+%29+%5Ccdot+%5Ccos+%5Cleft+%28+%5Cfrac%7Bx%7D%7B2%7D+%5Cright+%29%7D+%5C%5C%5C%5C%5C%5C+%5Cboxed%7B%5Cmathsf%7Bg%27%28x%29+%3D+20+%5Ccdot+%5Csin+%5Cleft+%28+%5Cfrac%7Bx%7D%7B2%7D+%5Cright+%29+%5Ccdot+%5Ccos+%5Cleft+%28+%5Cfrac%7Bx%7D%7B2%7D+%5Cright+%29%7D%7D "\\ \displaystyle \mathsf{g(x) = 20 \cdot \sin^2 \left ( \frac{x}{2} \right )} \\\\\\ \mathsf{g(x) = 20 \cdot \left [ \sin \left ( \frac{x}{2} \right ) \right ]^2} \\\\\\ \mathsf{g'(x) = 20 \cdot 2 \cdot \left [ \sin \left ( \frac{x}{2} \right ) \right ]^{2 - 1} \cdot \left [ \sin \left ( \frac{x}{2} \right ) \right ]'} \\\\\\ \mathsf{g'(x) = 20 \cdot 2 \cdot \left [ \sin \left ( \frac{x}{2} \right ) \right ]^1 \cdot \left [ \cos \left ( \frac{x}{2} \right ) \cdot \left ( \frac{x}{2} \right )' \right ]} \\\\\\ \mathsf{g'(x) = 20 \cdot 2 \cdot \sin \left ( \frac{x}{2} \right ) \cdot \left [ \cos \left ( \frac{x}{2} \right ) \cdot \left ( \frac{1}{2} \right ) \right ]} \\\\\\ \mathsf{g'(x) = 20 \cdot 2 \cdot \sin \left ( \frac{x}{2} \right ) \cdot \cos \left ( \frac{x}{2} \right ) \cdot \frac{1}{2}} \\\\\\ \mathsf{g'(x) = \frac{20 \cdot 2}{2} \cdot \sin \left ( \frac{x}{2} \right ) \cdot \cos \left ( \frac{x}{2} \right )} \\\\\\ \boxed{\mathsf{g'(x) = 20 \cdot \sin \left ( \frac{x}{2} \right ) \cdot \cos \left ( \frac{x}{2} \right )}}")
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