Let f(x)=sqrt(theta)sintheta=(theta)^(1/2)sintheta
differentiate with respect to theta using product rule
(df(x))/(d theta)=(d(theta)^(1/2))/(d theta)sintheta+theta^(1/2)(d(sintheta))/(d theta)
implies (df(x))/(d theta)=1/2theta^(1/2-1)sintheta+theta^(1/2)costheta
implies (df(x))/(d theta)=1/2theta^(-1/2)sintheta+sqrt(theta)costheta
implies (df(x))/(d theta)=1/2(1/theta^(1/2))sintheta+sqrt(theta)costheta
implies (df(x))/(d theta)=1/2(1/sqrt(theta))sintheta+sqrtthetacostheta
implies (df(x))/(d theta)=1/(2sqrttheta)sintheta+sqrtthetacostheta