Answer
$$\eqalign{
& f\left( {x + \Delta x} \right) = 4.998 \cr
& \root 4 \of {624} \approx 4.99799 \cr} $$
Work Step by Step
$$\eqalign{
& \root 4 \of {624} \cr
& {\text{Using }}f\left( x \right) = \root 4 \of x \cr
& f\left( x \right) = {x^{1/4}} \cr
& {\text{Differentiating}} \cr
& f'\left( x \right) = \frac{1}{4}{x^{ - 3/4}} \cr
& f'\left( x \right) = \frac{1}{{4\root 4 \of {{x^3}} }} \cr
& {\text{Using the formula}} \cr
& f\left( {x + \Delta x} \right) \approx f\left( x \right) + f'\left( x \right)dx \cr
& {\text{We can write,}} \cr
& f\left( {x + \Delta x} \right) \approx \root 4 \of x + \frac{1}{{4\root 4 \of {{x^3}} }}dx \cr
& f\left( {x + \Delta x} \right) = \root 4 \of {624} \cr
& {\text{Now, choosing }}x = 625{\text{ and }}dx = - 1,{\text{ and substituting }} \cr
& f\left( {x + \Delta x} \right) \approx \root 4 \of {625} + \frac{1}{{4\root 4 \of {{{\left( {625} \right)}^3}} }}\left( { - 1} \right) \cr
& f\left( {x + \Delta x} \right) = 5 - \frac{1}{{500}} \cr
& f\left( {x + \Delta x} \right) = 4.998 \cr
& \cr
& {\text{*Using a calculator to obtain }}\root 4 \of {624} \cr
& \root 4 \of {624} \approx 4.99799 \cr} $$
