Question

Qual a derivada de F(x) Ln(x)/raiz x

Answer 1

Regra do quociente:

$\boxed{(\frac{u}{v})'=\frac{u'v-uv'}{v^2}}$

Então, temos que:

$f(x)=\frac{lnx}{\sqrt{x}}\\\\ f'(x)=\frac{\frac{1}{x}(\sqrt{x})-lnx(\frac{1}{2\sqrt{x}})}{(\sqrt{x})^2}\\\\ f'(x)=\frac{\frac{\sqrt{x}}{x}-\frac{lnx}{2\sqrt{x}}}{x}\\\\ f'(x)=\frac{\frac{2\sqrt{x}\sqrt{x}-xlnx}{2x\sqrt{x}}}{x}\\\\ f'(x)=\frac{2x-xlnx}{2x^2\sqrt{x}}\\\\ \boxed{f'(x)=\frac{x(2-lnx)}{2x^2\sqrt{x}} }$