Step 1: Half-Angle Substitution:
Write $1+\sin5x=\left(\sin\frac{5x}2+\cos\frac{5x}2\right)^2$ and $\cos5x=\left(\cos\frac{5x}2-\sin\frac{5x}2\right)\left(\cos\frac{5x}2+\sin\frac{5x}2\right)$.
Step 2: Divide:
The ratio becomes $\dfrac{\cos\frac{5x}2+\sin\frac{5x}2}{\cos\frac{5x}2-\sin\frac{5x}2}=\dfrac{1+\tan\frac{5x}2}{1-\tan\frac{5x}2}=\tan\left(\frac\pi4+\frac{5x}2\right)$.
Step 3: Result:
So $y=\dfrac\pi2-\left(\dfrac\pi4+\dfrac{5x}2\right)=\dfrac\pi4-\dfrac{5x}{2}$ and $y'=-\dfrac52$. Option (D).
Final Answer:
Option (D).
\[ \boxed{\text{(D) } -\frac{5}{2}} \]