Find the second, third, and fourth derivatives of $y=2x^{3}$.

Solution Since $y$ is a power function, we use the Simple Power Rule and the Constant Multiple Rule to find each derivative. The first derivative is $$y^\prime =\dfrac{d}{dx}( 2x^{3}) = 2\cdot \dfrac{d}{dx}x^{3}=2\cdot 3x^{2}=6x^{2}$$

The next three derivatives are

$$y''=\dfrac{d^{2}}{dx^{2}}( 2x^{3}) =\dfrac{d}{dx}( 6x^{2}) =6\cdot \dfrac{d}{dx}x^{2}=6\cdot 2x=12x\\[5pt]$$

$$y'''=\dfrac{d^{3}}{dx^{3}}( 2x^{3}) =\dfrac{d}{dx}( 12x) =12$$

$$y^{(4)}=\dfrac{d^{4}}{dx^{4}}( 2x^{3}) =\dfrac{d}{dx}12=0$$