How do you differentiate $f (x) = - cos (\sqrt{\frac{1}{x^{2}}} - x)$ $f(x)=-cos(sqrt(1/(x^2))-x)$ 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

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

How do you differentiate f(x)=-cos(sqrt(1/(x^2))-x)f(x)=−cos(√1x2−x)f(x)=-cos(sqrt(1/(x^2))-x) using the chain rule?