Question

Differentiate `2^(cos^(2) x)` w.r.t cos² x.

Answer 1

Let u = `2^(cos^(2) x)`

Differentiate w. r. t., x

`(du)/(dx) = 2^(cos^(2) x)log^2(-2 cos x sin x)`

`(du)/(dx) = -2 cos x sin x  2^(cos^(2) x)log 2`

and again let v = cos² x

`(dv)/(dx) = -2 cos x sin x`

`(du)/(dx) = ((du)/(dx))/((dv)/(dx))`

= `(-2 cos x sin x  2^(cos^(2) x)log 2)/(-2 cos x sin x)`

∴ `(du)/(dv) = 2^(cos^(2) x)log 2`