Question

Cum se deriveaza asta?

Answer 1

$\\ \left(\sqrt[3]{x^2}\right)' => \\ \boxed{\frac{df\left(u\right)}{dx}=\frac{df}{du}\cdot \frac{du}{dx}} \\ f=\sqrt[3]{u},\:\:u=\left(x^2\right) \\ => \frac{1}{3u^{\frac{2}{3}}}\cdot \: 2x=> \frac{1}{3\left(x^2\right)^{\frac{2}{3}}}\cdot \:2x => Simplificare: \frac{2x}{3\left(x^2\right)^{\frac{2}{3}}} \\ \left(4x^2\sqrt{x}\right)' => \\ \boxed{ \left(f\cdot g\right)'=f\:'\cdot g+f\cdot g' } \\ f=x^2,\:g=\sqrt{x} => 4\left(\left(x^2\right)'\sqrt{x}+\left(\sqrt{x}\right)'x^2\right) \\ => 4\left(2x\sqrt{x}+\frac{x^{\frac{3}{2}}}{2}\right) \\ Deci: \:\ (\sqrt[3]{x^2}+4x^2\sqrt{x})' = \boxed{ \frac{2x}{3\left(x^2\right)^{\frac{2}{3}}}+4\left(2x\sqrt{x}+\frac{x^{\frac{3}{2}}}{2}\right) }$