Derivative of (t-1)/(t+1)

1. Apply the quotient rule, which is:

   $$\frac{d}{d t} \frac{f{\left(t \right)}}{g{\left(t \right)}} = \frac{- f{\left(t \right)} \frac{d}{d t} g{\left(t \right)} + g{\left(t \right)} \frac{d}{d t} f{\left(t \right)}}{g^{2}{\left(t \right)}}$$

   $f{\left(t \right)} = t - 1$ and $g{\left(t \right)} = t + 1$.

   To find $\frac{d}{d t} f{\left(t \right)}$:

   1. Differentiate $t - 1$ term by term:

      1. The derivative of the constant $-1$ is zero.

      2. Apply the power rule: $t$ goes to $1$

      The result is: $1$

   To find $\frac{d}{d t} g{\left(t \right)}$:

   1. Differentiate $t + 1$ term by term:

      1. The derivative of the constant $1$ is zero.

      2. Apply the power rule: $t$ goes to $1$

      The result is: $1$

   Now plug in to the quotient rule:

   $\frac{2}{\left(t + 1\right)^{2}}$

The answer is:

$$\frac{2}{\left(t + 1\right)^{2}}$$