Differentiating a Product in Two Ways

Find the derivative of \(F(v)=( 5v^{2}-v+1) ( v^{3}-1)\) in two ways:

1. By using the Product Rule.
2. By multiplying the factors of the function before finding its derivative.

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Solution

(a) \(F\) is the product of the two functions \(f(v) =5v^{2}-v+1\) and \(g( v) =v^{3}-1\). Using the Product Rule, we get \[ \begin{eqnarray*} F^\prime ( v) &=& ( 5v^{2}-v+1) \left[\dfrac{d}{dv}(v^{3}-1)\right]+\left[\dfrac{d}{dv}(5v^{2}-v+1)\right]( v^{3}-1)\\[5pt] &=&(5v^{2}-v+1) (3v^{2}) +(10v-1) (v^{3}-1)\\[5pt] &=&15v^{4}-3v^{3}+3v^{2}+10v^{4}-10v-v^{3}+1\\[5pt] &=&25v^{4}-4v^{3}+3v^{2}-10v+1 \end{eqnarray*} \]

(b) Here, we multiply the factors of \(F\) before differentiating. \[ F\left( v\right) =( 5v^{2}-v+1) ( v^{3}-1) =5v^{5}-v^{4}+v^{3}-5v^{2}+v-1 \]

Then \[ F' ( v) =25v^{4}-4v^{3}+3v^{2}-10v+1 \]