Gradient Descent in logistic regression Logistic and Linear Regression have different cost functions. But I don't get how the gradient descent in logistic regression is the same as Linear Regression. 

We get the Gradient Descent formula by deriving the Squared Error cost function. However in Logistic Regression we use a Logarithmic Cost function instead. I think I am lost here.
Gradient Descent is an universal method, you can us it with basically every loss function you can find in known ML algorithms.

In your case, you have only to derive the logarithmic cost function. You can find a detailed calculation at

https://math.stackexchange.com/questions/477207/derivative-of-cost-function-for-logistic-regression