Find the derivative of the function.
$y = (5 t - 1) (2 t - 4)^{- 1}$

Answer

Expert–verified

Hide Steps

Answer

So, the derivative of the function is $y^{'} = 5 / (2 t - 4) - 2 (5 t - 1) / (2 t - 4)^{2}$ .

Steps

Step 1 ：Let's denote $u = 5 t - 1$ and $v = (2 t - 4)^{- 1}$ .

Step 2 ：Then, we have $u^{'} = 5$ and $v^{'} = - 1 * (2 t - 4)^{- 2} * 2$ (using the chain rule), which simplifies to $v^{'} = - 2 / (2 t - 4)^{2}$ .

Step 3 ：Now, we can apply the product rule: $y^{'} = u^{'} v + u v^{'} = 5 * (2 t - 4)^{- 1} + (5 t - 1) * - 2 / (2 t - 4)^{2}$ .

Step 4 ：Simplifying this gives: $y^{'} = 5 / (2 t - 4) - 2 (5 t - 1) / (2 t - 4)^{2}$ .

Step 5 ：So, the derivative of the function is $y^{'} = 5 / (2 t - 4) - 2 (5 t - 1) / (2 t - 4)^{2}$ .