Dùng định nghĩa để tính đạo hàm của các hàm số sau:

d)  $f\left(x\right)=\sqrt[3]{x^2+1}$

d) Với  $x_0\in$ ℝ, ta có:

 $y^{\text{'}}\left(x_0\right)=\underset{x\to x_0}{\lim }\frac{f\left(x\right)-f\left(x_0\right)}{x-x_0}=\underset{x\to x_0}{\lim }\frac{\sqrt[3]{x^2+1}-\sqrt[3]{x_0^2+1}}{x-x_0}$

 $=\underset{x\to x_0}{\lim }\frac{\left(x+x_0\right)\left(\sqrt[3]{x^2+1}-\sqrt[3]{x_0^2+1}\right)}{\left(x-x_0\right)\left(x+x_0\right)}$

 $=\underset{x\to x_0}{\lim }\frac{\left(x+x_0\right)\left(\sqrt[3]{x^2+1}-\sqrt[3]{x_0^2+1}\right)}{x^2-x_0^2}$ $=\underset{x\to x_0}{\lim }\frac{\left(x+x_0\right)\left(\sqrt[3]{x^2+1}-\sqrt[3]{x_0^2+1}\right)}{\left(x^2+1\right)-\left(x_0^2+1\right)}$

 $=\underset{x\to x_0}{\lim }\frac{\left(x+x_0\right)\left(\sqrt[3]{x^2+1}-\sqrt[3]{x_0^2+1}\right)}{\left(\sqrt[3]{x^2+1}-\sqrt[3]{x_0^2+1}\right)\left[\sqrt[3]{{\left(x^2+1\right)}^2}+\sqrt[3]{x^2+1}\sqrt[3]{x_0^2+1}+\sqrt[3]{{\left(x_0^2+1\right)}^2}\right]}$

 $=\underset{x\to x_0}{\lim }\frac{x+x_0}{\sqrt[3]{{\left(x^2+1\right)}^2}+\sqrt[3]{x^2+1}\sqrt[3]{x_0^2+1}+\sqrt[3]{{\left(x_0^2+1\right)}^2}}=\underset{x\to x_0}{\lim }\frac{2x_0}{3\sqrt[3]{{\left(x_0^2+1\right)}^2}}$.

Vậy  $y^{\text{'}}\left(x\right)=\frac{2x}{3\sqrt[3]{{\left(x^2+1\right)}^2}}$ ( $x\in$  ℝ).