How do you differentiate #f(x)=-cos(sqrt(1/(x^2))-x)# using the chain rule?

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Answer 1 · 2016-10-21T04:14:43

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

Here you take the derivative of the outside, cos, and then the expression within the parenthesis.

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

#u'=-1x^-2-1=-x^-2-1=-1/x^2-1#

#g(u)=-cos(u)#

#g'(u)=-(-sin(u))=sin(u)#

#g'(u)*u'#

Substitute

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