Possible derivation:
d/dx(log(x cos(x)))
Simplify log(x cos(x)) using the identity log(a b) = log(a)+log(b):
= d/dx(log(x)+log(cos(x)))
Differentiate the sum term by term:
= d/dx(log(x))+d/dx(log(cos(x)))
The derivative of log(x) is 1/x:
= d/dx(log(cos(x)))+1/x
Using the chain rule, d/dx(log(cos(x))) = ( dlog(u))/( du) ( du)/( dx), where u = cos(x) and ( d)/( du)(log(u)) = 1/u:
= 1/x+d/dx(cos(x)) sec(x)
The derivative of cos(x) is -sin(x):
= 1/x+-sin(x) sec(x)
Simplify the expression:
Answer: | = 1/x-tan(x)
