Question

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

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

Answer 1

`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)`