Question

Using differential, find the approximate value of the following:
$(66{)}^{1/3}$

Answer 1

Let $f\left(x\right)=y=x^{1/3}$

Let $x=64$

$\Delta x=2$

$\therefore \frac{dy}{dx}=\frac{1}{3}{x}^{-2/3}=\frac{1}{3x^{2/3}}$

$\Rightarrow dy=\frac{1}{3x^{2/3}}dx$

$dy=\frac{1}{3(64{)}^{2/3}}\times 2=\frac{1}{3\left(16\right)}\times 2=\frac{1}{24}$

$dy=\frac{1}{24}$

$\Rightarrow y+dy=(x+\Delta x{)}^{1/3}$

$4+\frac{1}{24}=(64+2{)}^{1/3}$

$\Rightarrow (66{)}^{1/3}=4.0416$.