How do you differentiate #g(y) =(x^2 - 1) (x^2 - 2x + 1)^4 # using the product rule?

1 Answer
Steve M
Jul 26, 2017

# g'(y) = 2(x^2 - 2x + 1)^3( 5x^3-6x^2-3x+4 ) #

Explanation:

We have:

# g(y) = (x^2 - 1)(x^2 - 2x + 1)^4 #

In addition to the product rule we will also require the chain rule. Differentiating wrt #x# we get:

# g'(y) = (x^2 - 1)(d/dx(x^2 - 2x + 1)^4) + (d/dx(x^2 - 1))(x^2 - 2x + 1)^4 #

# " " = (x^2 - 1)(4(x^2 - 2x + 1)^3d/dx(x^2 - 2x + 1)) + (2x)(x^2 - 2x + 1)^4 #

# " " = (x^2 - 1)(4(x^2 - 2x + 1)^3(2x - 2)) + (2x)(x^2 - 2x + 1)^4 #

# " " = 2(x^2 - 2x + 1)^3(2(2x - 2)(x^2 - 1) + x(x^2 - 2x + 1)) #

# " " = 2(x^2 - 2x + 1)^3(2(2x^3-2x-2x^2+2) + (x^3 - 2x^2 + x)) #

# " " = 2(x^2 - 2x + 1)^3( 5x^3-6x^2-3x+4 ) #

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