To find the first derivative we must apply the chain rule. Our aim is to find dr/d(theta). We start by bringing the differential of what's inside the sine brackets outside and multiplying it by the differential of sine but keeping the same theta+sqrt(theta+1) for the whole sine differential. The differential we're bringing out is dr/d(theta) of theta+sqrt(theta+1) which is 1+1/2*(theta+1)-1/2 and the sine differentiates to cosine which becomes cos(theta+sqrt[theta+1]). Multiplying these both together gives us the answer by means of the chain rule of dr/d(theta)=(1+1/2*(theta+1)-1/2)cos(theta+sqrt[theta+1]). Simplifying it gives us the final answer of dr/d(theta)=(1+1/(2sqrt(theta+1)))*cos(theta+sqrt[theta+1]).