Question

For k(x)= (-x- 1)(x^2 + 3x - 1)(x+2), find the derivative of k(x) at the point = -2 using the product rule.

Answer 1

The derivative of that product of functions evaluated at the point x=-2 gives "-3"

Explanation:

Recall that the derivative of a product of two functions f(x) and g(x) is given by the formula:

(f*g)'= f' * g+ f * g'
`(2(-2)+3)\,(-(-2)^2-3(-2)-2)+((-2)^2+3(-2)-1)\,(-2(-2)-3)=`

So it would be convenient to reduce this product of three functions to a product of just 2, performing (-x-1)*(x+2) = - x^2 - 2x - x -2 = - x^2 -3x -2

therefore we need to find the derivative of x^2 + 3x -1, and the derivative of - x^2 -3x -2 to obtain the answer:

`(x^2 + 3x -1)'=2x+3\\ \\(- x^2- 3x -2)'=-2x-3`

Now, applying the product rule for those two trinomial functions, we get:

`(2x+3)\,(-x^2-3x-2)+(x^2+3x-1)\,(-2x-3)`

which at x = -2 becomes:

`(2x+3)\,(-x^2-3x-2)+(x^2+3x-1)\,(-2x-3)\\(2(-2)+3)\,(-(-2)^2-3(-2)-2)+((-2)^2+3(-2)-1)\,(-2(-2)-3)= -3`