Rewrite the expression as an algebraic expression in x.
cos(tan−1 x)

It's hard to type and hard to read the "inverse tangent" function, as you've seen (above).

So, use "arctan x" instead.

Then the problem becomes:  "differentiate cos (arctan x)."

You must apply first the rule for differentiating the cosine function, and next apply the rule for differentiating the arctan function:

(d/dx) cos (arctan x) = - sin (arctan x) * [1/(1+x^2)]

Physical stimulation can not be used as you carnt ask 1/2 a person a it would be painful