Letf(x)=x^x,x belongs to (0,∞) and letg(x) be inverse of f(x) ,then g’(x) must be

2086 Points
5 years ago
we know that f(g(x))= x
diff both sides using chain rule
f’(g(x))*g’(x)=1
or g’(x)= 1/f’(g(x))
now f’(x)= x^x(1+lnx)= f(x)*(1+lnx)
so f’(g(x))= f(g(x))*(1+lng(x)), but f(g(x))= x
so f’(g(x))= x(1+lng(x))
so that g’(x)= 1/f’(g(x))= 1/x(1+lng(x))
option c is correct.
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