How do you differentiate # f(x)=ln(1/sqrt(xe^x-x))# using the chain rule.?

1 Answer
Guilherme N.
Dec 23, 2015

Using two power rules, we can simplify the expression and then proceed to derivate it more simply.

Explanation:

Two power rules to be remembered:

  • #a^(-n)=1/(a^n)#
  • #a^(m/n)=root(n)(a^m)#

Rewriting the expression: #f(x) = (xe^x-x)^(-1/2)#, then.

Using the chain rule, which states that #(dy)/(dx)=(dy)/(du)(du)/(dx)#, let's rename #u=xe^x-x# and derivate it:

#(df(x))/dx=-(1/2)u^(-3/2)(e^x+xe^x-1)=-(e^x+xe^x-1)/(2(xe^x-x)^(3/2))#

Just organizing:

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

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