Question

Решите, пожалуйста, номер 4.

Answer 1

найти $\mathtt{f'(x)}$, если $\mathtt{f(x)=\sqrt{x^2+1}-\ln\sqrt[4]{x^2+1}}$

по правилу производной разности,

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