find the derrivitiv of Y= X^6.(X-1)^10.(X+2)^8. Express in factored form.
There are only two variables here, X and Y.
You have the product of three terms on the rhs, so use the product rule, twice.
Let Y = UV, where U = X^6, and V = (X-1)^10.(X+2)^8
Then Y' = UV' + VY' = x^6.V' + V.6x^5
Let V = WZ, where W = (X-1)^10, and Z = (X+2)^8
Then V' = WZ' + ZW' = (X-1)^10{8(X+2)^7} + (X+2)^8{10(X-1)^9}
Substituting back for V' into Y',
Y' = x^6[(X-1)^10{8(X+2)^7} + (X+2)^8{10(X-1)^9}] + [(X-1)^10.(X+2)^8].6X^5
Y' = X^5.(X-1)^9.(X+2)^7{X[(X-1).8 + (X+2).10] + 6[(X-1)(X+2)]}
Y' = X^5.(X-1)^9.(X+2)^7{[8X^2 - 8X + 10X^2 + 20X] + 6{X^2 +X - 2]}
Y' = X^5.(X-1)^9.(X+2)^7{24X^2 + 18X - 12}
Y' = 6X^5.(X-1)^9.(X+2)^7{4X^2 + 3X - 2}