How do you use the Product Rule to find the derivative of #y = x(x^2 - 2x + 1)^4#?

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Answer 1 · 2015-08-01T00:03:31

#y^' = (9x-1) * (x-1)^7#

The product rule allows you to differentiate functions that can be written as the product of two other functions

#y = f(x) * g(x)#

by using this formula

#color(blue)(d/dx(y) = f^'(x) * g(x) + f(x) * g^'(x)#

In your case, you can write #y# as

#y = x * (x^2 - 2x + 1)^4 = x * [(x-1)^2]^4 = x * (x-1)^8#

This means that its derivative can be found by using the product rule

#y^' = [d/dx(x)] * (x-1)^8 + x * d/dx(x-1)^8#

#y^' = 1 * (x-1)^8 + x * 8 * (x-1)^7#

#y^' = (x-1)^7(x-1 + 8x) = color(green)( (9x-1) * (x-1)^7)#