Question

y=
$|x { }^{3} - 1 | {?}^{5}$
qual a derivada ​

Answer 1

$y=\left|x^3-1\right|^5\Longrightarrow y=\left(x^3-1\right)^4\left|x^3-1\right|$

$\Longrightarrow \dfrac{\mathrm{d}y}{\mathrm{d}x}=\dfrac{\mathrm{d}}{\mathrm{d}x}\left(\left(x^3-1\right)^4\left|x^3-1\right|\right)$

$\Longrightarrow \dfrac{\mathrm{d}y}{\mathrm{d}x}=|x^3-1|\dfrac{\mathrm{d}}{\mathrm{d}x}\left(\left(x^3-1\right)^4\right)+\left(x^3-1\right)^4\dfrac{\mathrm{d}}{\mathrm{d}x}\left(|x^3-1|\right)$

$\Longrightarrow \dfrac{\mathrm{d}y}{\mathrm{d}x}=\left|x^3-1\right|4\left(x^3-1\right)^3\left(3x^2\right)+\left(x^3-1\right)^4\dfrac{x^3-1}{\left|x^3-1\right|}\left(3x^2\right)$

$\Longrightarrow \dfrac{\mathrm{d}y}{\mathrm{d}x}=12x^2\left(x^3-1\right)^3\left|x^3-1\right|+\dfrac{3x^2\left(x^3-1\right)^5}{\left|x^3-1\right|}$

$\Longrightarrow \boxed{\dfrac{\mathrm{d}y}{\mathrm{d}x}=15x^2\left(x^3-1\right)^3\left|x^3-1\right|}$