Answer
$$dy = 0.6{\text{ and }}\Delta y = 0.661$$
Work Step by Step
$$\eqalign{
& {\text{Let the function }}f\left( x \right) = {x^3} - 6x,{\text{ }} \cr
& x{\text{ - value: }}x = 2,{\text{ }} \cr
& {\text{Differential of }}x:{\text{ }}\Delta x = dx = 0.1 \cr
& \cr
& {\text{Differentiating:}} \cr
& f\left( x \right) = {x^3} - 6x \cr
& f'\left( x \right) = 3{x^2} - 6 \cr
& {\text{The differential is }} \cr
& dy = f'\left( x \right)dx \cr
& dy = \left( {3{x^2} - 6} \right)dx \cr
& {\text{Substituting }}x = 2{\text{ and }}dx = 0.1 \cr
& dy = \left( {3{{\left( 2 \right)}^2} - 6} \right)\left( {0.1} \right) \cr
& dy = 0.6 \cr
& \cr
& {\text{Now}},{\text{ using }}\Delta x = 0.1,{\text{ the change in }}y{\text{ is}} \cr
& \Delta y = f\left( {x + \Delta x} \right) - f\left( x \right) \cr
& \Delta y = f\left( {2 + 0.1} \right) - f\left( 2 \right) \cr
& \Delta y = f\left( {2.1} \right) - f\left( 2 \right) \cr
& \Delta y = {\left( {2.1} \right)^3} - 6\left( {2.1} \right) - \left[ {{{\left( 2 \right)}^3} - 6\left( 2 \right)} \right] \cr
& \Delta y = 0.661 \cr
& \cr
& dy = 0.6{\text{ and }}\Delta y = 0.661 \cr} $$
