Question

Como calcular a derivada f(x)= 4x^1/2 +5x^-1/2 ?

Answer 1

$\boxed{\boxed{(x^{n})' = n*x^{(n-1)}}}$

aplicando isso:
$f(x)= 4x^{\frac{1}{2}} + 5x^{\frac{-1}{2}} \\\\ \text{a derivada de uma soma eh a soma das derivadas}\\\\ f'(x)= (4x^{\frac{1}{2}})' + (5x^{\frac{-1}{2}})'\\\\\ \text{mantem a constante e deriva a variavel}\\\\\ f'(x)=4(x^{\frac{1}{2}})' + 5(x^{\frac{-1}{2}})'\\\\ f'(x)= 4*( \frac{1}{2}*x^{ \frac{1}{2}-1 } ) + 5*( \frac{-1}{2}* x^{\frac{-1}{2}-1 })\\\\ f'(x)=4*( \frac{1}{2}*x^{ \frac{-1}{2} } )+ 5*( \frac{-1}{2}*x^{ \frac{-3}{2} } )$


$f'(x)= \frac{4}{2}*x^{ \frac{-1}{2} } - \frac{5}{2}*x^{ \frac{-3}{2} } \\\\ f'(x)= 2*x^{ \frac{-1}{2} } - \frac{5}{2}* x^\frac{-3}{2} \\\\f'(x)= \frac{2}{x^{ \frac{1}{2} }} - \frac{5}{2x^{ \frac{3}{2} }} \\\\f'(x) = \frac{2}{ \sqrt{x} }- \frac{5}{ \sqrt[3]{x^2} }$