Answer
$${\left( {1 + \Delta x} \right)^4} \approx 1 + 4\Delta x$$
Work Step by Step
$$\eqalign{
& f\left( x \right) = {x^4};{\text{ }}{\left( {1 + \Delta x} \right)^4} \approx 1 + 4\Delta x,{\text{ }}{x_0} = 1,{\text{ where }}\Delta x = x - 1 \cr
& {\text{Calculate }}f'\left( x \right) \cr
& f'\left( x \right) = 4{x^3} \cr
& {\text{Evaluate }}f\left( {{x_0}} \right){\text{ and }}f'\left( {{x_0}} \right){\text{ }} \cr
& f\left( {{x_0}} \right) = {\left( 1 \right)^4} = 1 \cr
& f'\left( {{x_0}} \right) = 4{\left( 1 \right)^3} = 4 \cr
& {\text{Apply }}f\left( {{x_0} + \Delta x} \right) \approx f\left( {{x_0}} \right) + f'\left( {{x_0}} \right)\Delta x{\text{ and substitute the known values}} \cr
& {\text{ }}{\left( {1 + \Delta x} \right)^4} \approx 1 + 4\Delta x \cr} $$
