Single Variable Calculus, Chapter 3, 3.3, Section 3.3, Problem 32

Differentiate $\displaystyle y = \frac{t}{(t - 1)^2}$



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\begin{equation}
\begin{aligned}

y' =& \frac{(t^2 - 2t + 1) \displaystyle \frac{d}{dt} (t) - \left[ (t) \frac{d}{dt} (t^2 - 2t + 1) \right] }{(t - 1)^4}
&& \text{Apply Quotient Rule}
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y' =& \frac{(t^2 - 2t + 1)(1) - [(t)(2t - 2)]}{(t - 1)^4}
&& \text{Expand the equation}
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y' =& \frac{t^2 - \cancel{2t} + 1 - 2t^2 + \cancel{2t}}{(t - 1)^4}
&& \text{Combine like terms}
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y' =& \frac{1 - t^2}{(t - 1)^4}
&& \text{Get the factor of numerator and denominator}
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y' =& \frac{-1(t + 1) \cancel{(t - 1)} }{(t - 1)^3 \cancel{(t - 1)}}
&& \text{Cancel out like terms}
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y' =& \frac{-1 (t + 1)}{(t - 1)^3} \text{ or } \frac{-t - 1}{(t - 1)^3}

\end{aligned}
\end{equation}
$