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

1 Answer
Lee E.
Oct 5, 2014

The product rule states that (hfg)'=h'fg+hf'g+hfg'.

So, (d/dx(5-x))(x-3)(2-3x)+(5-x)(d/dx(x-3))(2-3x)+(5-x)(x-3)(d/dx(2-3x))

The derivative of 5-x is -1 since the constant has a derivative of o and the derivative of -x is -1 from (1)*-1x^(1-1) giving -1x^0 or -1.

The derivative of x-3 is 1 as above.

The derivative of 2-3x is -3 from 1*-3x^(1-1)=-3^0

Substituting back we get =(-1)((x-3))((2-3x))+((5-x))(1)((2-3x))+((5-x))((x-3))(-3)

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

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

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