Differentiating a stochastic integral

Question

How do i differentiate the following stochatic integral?

$$\frac {d}{dW_t} \int_{0}^t \frac{1}{1-u} dW_u$$

My guess is

$$\frac {d}{dW_t} \int_{0}^t \frac{1}{1-u} dW_u = \left.\frac {1}{1-u} \right\vert_{u=0}^{u=t}=\frac {1}{1-t} - 1$$

but ive seen it solved without the lower limit and just be

$$\frac {d}{dW_t} \int_{0}^t \frac{1}{1-u} dW_u = \frac {1} {1-t}$$

Answer 1

Well if you do the standard FTC you would just get the upper limit, so this makes intuitive sense.