Question

Determinação de derivada,(2x³+2x)⁴

Answer 1

$f(x)=(2x^3+2x)^4$

Podemos enxergar $f$ como uma função composta:

$\left\{ \!\begin{array}{l} f(x)=\big[g(x)\big]^4\\\\ g(x)=2x^3+2x \end{array} \right.$

Então, vamos derivar $f$ usando a Regra da Cadeia:

$\dfrac{df}{dx}(x)=\dfrac{d}{dx}\left(\big[g(x)\big]^4 \right )\\\\\\ \dfrac{df}{dx}(x)=4\big[g(x)\big]^{4-1}\cdot \dfrac{d}{dx}\big[g(x)\big]\\\\\\ \dfrac{df}{dx}(x)=4\big[g(x)\big]^{3}\cdot \dfrac{d}{dx}\big[g(x)\big]\\\\\\ \dfrac{df}{dx}(x)=4\,(2x^3+2x)^{3}\cdot \dfrac{d}{dx}(2x^3+2x)\\\\\\ \boxed{\begin{array}{c} \dfrac{df}{dx}(x)=4\,(2x^3+2x)^{3}\cdot (6x^2+2) \end{array}}$