How do you differentiate #f(x) =(1-x)/(x^3-x)# using the quotient rule?

1 Answer
Jim H
Feb 7, 2017

see below

Explanation:

Using the quotient rule.

#f'(x) = ((-1)(x^3-x)-(1-x)(3x^2-1))/(x^3-x)^2#

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

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

# = (2x^2-x-1)/(x^2(x-1)(x+1)^2)#

# = ((2x+1)(x-1))/(x^2(x-1)(x+1)^2)#

# = (2x+1)/(x^2+x)^2#

Simplifying first

#f(x) = (1-x)/(x^3-x) = (-(x-1))/(x(x+1)(x-1))#

# = (-1)/(x^2+x) = -(x^2+x)^-1#

#f'(x) = 1(x^2+x)^-2 (2x+1)# #" "# (Chain rule)

# = (2x+1)/(x^2+x)^2#

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