Answer
$$y'=-\frac{2t+1}{2(t^2+t)\sqrt{t^2+t}}$$
Work Step by Step
$$y=\sqrt{\frac{1}{t(t+1)}}=\sqrt{\frac{1}{t^2+t}}$$
Logarithmic differentiation helps to simplify the differentiation of otherwise complex functions by using the algebraic properties of logarithm. In detail,
1) Take the natural logarithm of both sides and simplify:
$$\ln y=\ln\sqrt{\frac{1}{t^2+t}}=\ln\Big(\frac{1}{\sqrt{t^2+t}}\Big)$$
Recall that $\ln(A/ B)=\ln A-\ln B$: $$\ln y=\ln1-\ln\sqrt{t^2+t}=0-\ln\sqrt{t^2+t}$$ $$\ln y =-\ln\sqrt{t^2+t}$$
2) Then we take the derivative of both sides with respect to $x$ and remember that $(\ln x)'=1/x$:
$$\frac{1}{y}\times y'=-\frac{1}{\sqrt{t^2+t}}(\sqrt{t^2+t})'$$ $$\frac{y'}{y}=-\frac{1}{\sqrt{t^2+t}}\times\frac{1}{2\sqrt{t^2+t}}(t^2+t)'$$ $$\frac{y'}{y}=-\frac{(t^2+t)'}{2(t^2+t)}$$ $$\frac{y'}{y}=-\frac{2t+1}{2(t^2+t)}$$
3) Solve for $y'$: $$y'=-\frac{2t+1}{2(t^2+t)}\times y$$
4) Finally, substitute for $y=\sqrt{\frac{1}{t^2+t}}$
$$y'=-\frac{2t+1}{2(t^2+t)}\times\sqrt{\frac{1}{t^2+t}}=-\frac{2t+1}{2(t^2+t)\sqrt{t^2+t}}$$
