find the derrivitiv of Y= X^6(X-1)^10(X+2)^8. Express in factored form.

Question

i know you have to use the power rule but i have no idea how with 3 variables.

Grade 12 calculus question.

Answer 1

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}