Question

So para os ninjas da engenharia valendo 50 pontos..... Calculem essa derivada
\frac{(4 x^{3}-6x).(4 x^{3} +4 x^{2} -1)-( x^{4} -3 x^{2} +1).(12 x^{2}+8x) }{(4 x^{3}+4 x^{2} -1) }

Answer 1

$y=\frac{(4 x^{3}-6x)*(4 x^{3} +4 x^{2} -1)-( x^{4} -3 x^{2} +1)*(12 x^{2}+8x) }{(4 x^{3}+4 x^{2} -1) }\\\\y= \frac{A*B-C*D}{B} \\\\\boxed{\boxed{y=A -\frac{C*D}{B} }}$

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$y'=A'- \frac{[C*D]' *B -[C*D]*B'}{B^2} \\\\\boxed{\boxed{y' = A'- \frac{[C'*D+C*D'] *B -[C*D]*B'}{B^2}}}$

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$\bmatrix[ A=4x^3-6x \\A'=12x^2-6 \\\\B=4x^3+4x^2-1 \\B'=12x^2+8x=D \\\\C=x^4-3x^2+1 \\C'=4x^3-6x=A \\\\D=12x^2+8x \\D'=24x+8\end$

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$y' = A'- \frac{[C'*D+C*D'] *B -[C*D]*D}{B^2}}\\\\\boxed{\boxed{y'=A'- \frac{[C'*D+C*D'] }{B}+ C*\left(\frac{D}{B}\right)^2 }}\\\\\.$