Answer
$$y'=\Big(\frac{\ln\sin\theta}{2\sqrt\theta}+\sqrt\theta\cot\theta\Big)(\sin\theta)^{\sqrt\theta}$$
Work Step by Step
$$y=(\sin\theta)^{\sqrt\theta}$$
Take the natural logarithm of both sides: $$\ln y=\ln(\sin\theta)^{\sqrt\theta}$$ $$\ln y=\sqrt\theta(\ln\sin\theta)$$
Now take the derivative of both sides, and remember that $(\ln \theta)'=1/\theta$:
$$\frac{1}{y}\times y'=(\sqrt\theta)'(\ln\sin\theta)+\sqrt\theta(\ln\sin\theta)'$$ $$\frac{1}{y}\times y'=\frac{\ln\sin\theta}{2\sqrt\theta}+\sqrt{\theta}\Big(\frac{1}{\sin\theta}\times(\sin\theta)'\Big)$$ $$\frac{y'}{y}=\frac{\ln\sin\theta}{2\sqrt\theta}+\sqrt\theta\Big(\frac{\cos\theta}{\sin\theta}\Big) $$ $$\frac{y'}{y}=\frac{\ln\sin\theta}{2\sqrt\theta}+\sqrt\theta\cot\theta$$
Find $y'$: $$y'=\Big(\frac{\ln\sin\theta}{2\sqrt\theta}+\sqrt\theta\cot\theta\Big)y$$ $$y'=\Big(\frac{\ln\sin\theta}{2\sqrt\theta}+\sqrt\theta\cot\theta\Big)(\sin\theta)^{\sqrt\theta}$$
