Differentiating a Constant Times a Power Function

Find the derivative of each power function:

1. \(f( x) =5x^{3}\)
2. \(g( u) =-\dfrac{1}{2}u^{2} \)
3. \(u( x) =\pi ^{4}x^{3}\)

Solution Notice that each of these functions involves the product of a constant and a power function. So, we use the Constant Multiple Rule followed by the Simple Power Rule.

1. \(f( x) =5\cdot x^{3},\) so \(f^\prime ( x) =5\left[ \dfrac{d}{dx}x^{3}\right] =5\cdot 3x^{2}=15x^{2}\)
2. \(g( u) =-\dfrac{1}{2}\cdot u^{2}\), so \(g^\prime ( u) =-\dfrac{1}{2}\cdot \dfrac{d}{du}u^{2}=-\dfrac{1}{2}\cdot 2u^{1}=-u\)
3. \(u( x) =\pi ^{4}x^{3}, \)so \(u' ( x)\underset{\underset{\color{#0066A7}{\pi~\text{is a constant}}}{\color{#0066A7}{{\uparrow }}}}{=} \pi ^{4}\cdot \dfrac{d}{dx}x^{3}=\pi ^{4}\cdot 3x^{2}=3\pi ^{4}x^{2}\)