Question

Por favor derivar essa função passo a passo:

f (x) = 2 cos (x²) × sen (2x).

pontos para a melhor resposta.

Answer 1

$2cos~(x^{2})=g(x)\\sen~(2x)=h(x)$

Logo:

$f(x)=g(x)*h(x)~~~~\therefore~~~~\boxed{\boxed{f'(x)=f'(x)*g(x)+g'(x)*f(x)}}$
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Achando a derivada de g(x):

$g'(x)=\dfrac{d~2}{dx}*cos~(x^{2})+\dfrac{d~cos~(x^{2})}{dx}*2\\\\\\g'(x)=0*cos~(x^{2})+[-sen~(x^{2})]*2x*2\\\\g'(x)=-4x*sen~(x^{2})$

Achando a derivada de h(x):

$h'(x)=\dfrac{d~sen~(2x)}{dx}\\\\\\h'(x)=cos~(2x)*1*2x^{1-1}\\\\h'(x)=cos~(2x)*2\\\\h'(x)=2*cos~(2x)$
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$f'(x)=g'(x)*h(x)+h'(x)*g(x)\\f'(x)=-4x*sen~(x^{2})*sen~(2x)+2*cos~(2x)*2cos~(x^{2})\\f'(x)=-4x.sen~(x^{2}).sen~(2x)+4cos~(x^{2}).cos~(2x)\\\\\boxed{\boxed{f'(x)=4cos(x^{2}).cos(2x)-4x.sen(x^{2}).sen(2x)}}$