Question

qual a derivada de f(x)=raiz de (x^3-x^2)

Answer 1

considerando que f é uam função composta:
$f(x)= \sqrt{x^3-x^2}\ \ |\ \ f(u)= \sqrt{u} \ \ \ \ |\ \ \ \ u(x)=x^3-x^2$
temos:
$(f\circ u )(x)=\sqrt{x^3-x^2}$
pela regra da cadeia:
$\displaystyle \frac{df}{dx}=\frac{df}{du}.\frac{du}{dx}\\\\ \frac{df}{dx}=\frac{d}{du}\sqrt{u}.\frac{d}{dx}(x^3-x^2)$
agora só derivar e multiplicar:
$\displaystyle \frac{d}{du} \sqrt{u} =\frac{1}{2}u^{\frac{1}{2}-1}=\frac{1}{2u^{\frac{1}{2}}}=\frac{1}{2\sqrt{u}}\\\\ \frac{d}{dx}(x^3-x^2)=3x^2-2x\\\\ \\ \frac{df}{dx}=\frac{1}{2\sqrt{u}}.3x^2-2x=\frac{3x^2-2x}{2\sqrt{u}}\ \ \ \ \ |u=x^3-x^2\ |\\\\ \frac{df}{dx}=\frac{3x^2-2x}{2\sqrt{x^3-x^2}}$