Find the derivative

The derivate of   $$y=\dfrac{t}{\sqrt{(ct^2-1)}} = t*\left( ct^2-1\right)^{-\frac{1}{2}}$$

Product Rule: $$y^\prime=t * \left( \frac{d(ct^2-1)^{-\frac{1}{2}}}{dt} \right)+\frac{d(t)}{dt}*(ct^2-1)^{-\frac{1}{2}}$$

$$y^\prime=t *(-\frac{1}{\not{2}})(ct^2-1)^{-\frac{1}{2}-1}}*\not{2}ct + 1*(ct^2-1)^{-\frac{1}{2}}$$

$$y^\prime=-ct^2(ct^2-1)^{-\frac{3}{2}}} + (ct^2-1)^{-\frac{1}{2}}$$

$$y^\prime=\dfrac{1}{\sqrt{(ct^2-1)}} -\dfrac{ct^2}{ \left( \sqrt{(ct^2-1)} \right)^3 }$$

$$\boxed{y^\prime=\dfrac{1}{\sqrt{(ct^2-1)}} \left( 1-\dfrac{ct^2}{ct^2-1} \right)}$$

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