How do you use the chain rule to differentiate #f(x)=sin(1/(x^2+1))#?

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Answer 1 · 2018-03-26T11:45:30

See below

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

#f(x)=Sin(u)#

#f'(x)=Cos(u) times u'#

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

(Substitute again to include the inner function)

#v=x^2+1#
#v'=2x#

#u'=-1(v')(v)^-2=-1(2x)(x^2+1)^-2=(-2x)/(x^2+1)^2#

Hence:

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