In general for x near r,
[tex]f'(r) \approx \dfrac{f(x) - f(r)}{x - r}[/tex]
[tex]f(x) = f(r) + (x-r) f'(r)[/tex]
[tex]g(x) = (1+x)^{\frac 1 3}[/tex]
[tex]g'(x) = \frac 1 3 (1+x)^{-\frac{2}{3}}[/tex]
[tex]g'(r) = \frac 1 3 (1+r)^{-\frac{2}{3}}[/tex]
Near [tex]x=r[/tex] we get a linear approximation
[tex]g(x) \approx g(r) + (x-r) g'(r) = (1+r)^{\frac 1 3} + \dfrac{x-r}{3} (1+r)^{-\frac{2}{3}}[/tex]
That's messy. Typically we're asked for the approximation near a given point, often [tex]x=0[/tex]. We set [tex]r=0[/tex].
Near [tex]x=0[/tex]
[tex](1+x)^{\frac 1 3}\approx 1 + \dfrac{x}{3}[/tex]
