Question

How would I answer this question?
Differentiate the function f(x)=sqrt(cosxsinx)

Answer 1

simplify it ...

f(x) = √(sinxcosx)
= √( (1/2)(sin(2x) )
= 1/√2 (sin(2x))^(1/2)

f ' (x) = (1/√2)(1/2)(sin(2x))^(-1/2)(2cos(2x))
= (1/√2) (cos 2x)/(sin 2x)
= 1/√2 cot (2x)

or , the way it stands
f(x) = (sinxcosx)^(1/2)
f ' (x) = (1/2)(sinxcox)^(-1/2) (sinx(-sinx) + cosxcosx)
= (1/2)(cos^2 x - sin^2 x)/√(sinxcosx)
= cos(2x)/(2√(sinxcosx)