16. Hình bị mất khúc trên nên ko rõ đề
17.
\(y'=\dfrac{1}{2}\left(x^2+3\right)^{\dfrac{1}{2}-1}\left(x^2+3\right)'=\dfrac{1}{2}\left(x^2+3\right)^{-\dfrac{1}{2}}.2x=x\left(x^2+3\right)^{\dfrac{1}{2}}\)
18.
\(f\left(x\right)=\left(x^2+x+1\right)^{\dfrac{1}{3}}\Rightarrow f'\left(x\right)=\dfrac{1}{3}\left(2x+1\right)\left(x^2+x+1\right)^{-\dfrac{2}{3}}\)
\(\Rightarrow f'\left(0\right)=\dfrac{1}{3}.1.1=\dfrac{1}{3}\)
19.
\(f\left(x\right)=\left(\dfrac{x-1}{x+1}\right)^{\dfrac{1}{5}}\Rightarrow f'\left(x\right)=\dfrac{1}{5}.\left(\dfrac{x-1}{x+1}\right)^{-\dfrac{4}{5}}\left(\dfrac{2}{x+1}\right)^2\)
\(\Rightarrow f'\left(0\right)=\dfrac{1}{5}.1.4=\dfrac{4}{5}\)
20.
\(f\left(x\right)=\left(1+2sin2x\right)^{\dfrac{1}{3}}\Rightarrow f'\left(x\right)=\dfrac{1}{3}\left(1+2sin2x\right)^{-\dfrac{2}{3}}.4cos2x\)
\(\Rightarrow f'\left(0\right)=\dfrac{1}{3}.1.4=\dfrac{4}{3}\)
21.
\(f'\left(x\right)=\dfrac{1}{2}\left(x^5+x^3+1\right)^{-\dfrac{1}{2}}\left(5x^4+3x^2\right)=\dfrac{5x^4+3x^2}{2\sqrt{x^5+x^3+1}}\ge0;\forall x\)
\(\Rightarrow f\left(x\right)\) đồng biến
\(\Rightarrow\max\limits_{\left[1;3\right]}f\left(x\right)=f\left(3\right)=\sqrt{271}\)
