What is the derivative of the following function? g(x)=sqrt(x^(sin(x)))

g(x)=sqrt(x^(sin(x)))

1 Answer
Mar 18, 2018

#f' (x) = ((cos(x)ln(x)/2)+(sin(x)/(2x) ))*e^(((sin(x)*ln(x))/2)=((cos(x)ln(x)/2)+(sin(x)/(2x) ))*x^(sin(x)/2)#

Explanation:

We know that : #sqrt(x) =x^(1/2)#
We can simplify : #f(x) = x^(sin(x)/2)#
We also know that : #u(x) =e^ln(u(x))# AND #ln(a^b) =b*ln(a)#
So : #f(x) =e^(((sin(x)*ln(x))/2)#
And finally We have : #f(x) =e^u #<=> #f'(x) = u'*e^u#, and also #(u*v)'= u'v+v'u#
We have so : #f' (x) = ((cos(x)ln(x)/2)+(sin(x)/(2x) ))*e^(((sin(x)*ln(x))/2)=((cos(x)ln(x)/2)+(sin(x)/(2x) ))*x^(sin(x)/2)#