# How do you apply the product rule repeatedly to find the derivative of f(x) = (x - 3)(2 - 3x)(5 - x) ?

Oct 5, 2014

The product rule states that $\left(h f g\right) ' = h ' f g + h f ' g + h f g '$.

So, $\left(\frac{d}{\mathrm{dx}} \left(5 - x\right)\right) \left(x - 3\right) \left(2 - 3 x\right) + \left(5 - x\right) \left(\frac{d}{\mathrm{dx}} \left(x - 3\right)\right) \left(2 - 3 x\right) + \left(5 - x\right) \left(x - 3\right) \left(\frac{d}{\mathrm{dx}} \left(2 - 3 x\right)\right)$

The derivative of $5 - x$ is $- 1$ since the constant has a derivative of o and the derivative of $- x$ is $- 1$ from $\left(1\right) \cdot - 1 {x}^{1 - 1}$ giving $- 1 {x}^{0} \mathmr{and} - 1.$

The derivative of $x - 3$ is 1 as above.

The derivative of $2 - 3 x$ is -3 from $1 \cdot - 3 {x}^{1 - 1} = - {3}^{0}$

Substituting back we get $= \left(- 1\right) \left(\left(x - 3\right)\right) \left(\left(2 - 3 x\right)\right) + \left(\left(5 - x\right)\right) \left(1\right) \left(\left(2 - 3 x\right)\right) + \left(\left(5 - x\right)\right) \left(\left(x - 3\right)\right) \left(- 3\right)$

Using FOIL we get $3 {x}^{2} - 11 x + 6 + 3 {x}^{2} - 17 x + 10 + 3 {x}^{2} - 24 x + 45.$

Collecting like terms we get our derivative; $9 {x}^{2} - 52 x + 61.$