Question

Derive e simplifique a função a seguir, mostrando todas as etapas: y= $x^{6}.(2.x-5.lnx)$e $f(x)=(x^{4}-7.x+2).secx$

Answer 1

$\boxed{f(x) = x^6 *(2x-5ln(x))}$

utilizando a regra do produto
$\boxed{(U*V)' = U'*V + U *V'}$

temos
$u = x^6\\\\u' = 6x^5$

e

$v = 2x-5*ln(x)\\\\v' = 2 -5* \frac{1}{x}\\\\\ v' = 2- \frac{5}{x} = \frac{2x-5}{x}$

agora colocando na derivada
$6x^5 * (2x-5ln(x) ) + x^6*( \frac{2x-5}{x}) \\\\12x^6 - 30x^5ln(x) + x^5(2x-5)\\\\12x^6-30x^5 ln(x)+2x^6-5x^5\\\\14x^6-5x^5-30x^5ln(x)\\\\\boxed{f'(x)=x^5(14x-5-30ln(x))}$
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
$\boxed{f(x)=(x^4-7x+2)*sec(x)}$

utilizando a regra do produto novamente
$U = x^4-7x+2\\\\U'= 4x^3-7\\\\\\ V=sec(x)\\\\V' = sec(x)*tg(x)$

aplicando
$(4x^3-7) * sec(x) + (x^4+7x+2)*(sec(x) * tg(x))\\\\\\\boxed{f'(x)=sec(x) * [(4x^3-7) +(x^4+7x+2)*tg(x)]}$

só coloquei sec(x) em evidencia