Question

Determine as derivas abaixo:

$D) y= \frac{ x^{3}-2 }{ \sqrt{x} }$

$F) y= \frac{8-z+3 z^{2} }{2-9z}$[tex]

Answer 1

D )

Sem usar a regra do quociente é possível simplificar antes de derivar.

$f(x)= \frac{x^3-2}{ \sqrt{x} } \\ \\f(x)= \frac{x^3-2}{x^{ \frac{1}{2} }} \\ \\f(x)=(x^3-2)x^{ -\frac{1}{2} }\\ \\f(x)=x^{ \frac{5}{2} }-2x^{- \frac{1}{2} }$

$f'(x)= \frac{5}{2}x^{ \frac{3}{2} }+1x^{- \frac{3}{2} }$

$\boxed{\boxed{f'(x)= \frac{5 \sqrt{x^3} }{2}+ \frac{1}{ \sqrt{x^3} }}}$

F ) Usando a regra do quociente.

$f(z)= \frac{8-z+3z^2}{2-9z}$


$u=8-z+3z^2~~~~~~~~~~~~u'=6z-1\\v=2-9z~~~~~~~~~~~~~~~~~~v'=-9~\\\\v^2=(2-9z)^2$

$f'(z)= \frac{u'.v-u.v'}{v^2}$

$\boxed{\boxed{f'(z)= \frac{(6z-1)(2-9z)-(8-z+3z^2)(-9)}{(2-9z)^2} }}$