Preheat the oven to 350°F (175°C) and grease an 8x8-inch baking dish.
In a saucepan, melt the butter over low heat. Remove from heat and stir in 1 cup sugar, 2 eggs, and 1 teaspoon vanilla.
Mix in cocoa powder, flour, salt, and baking powder until smooth.
Pour 3/4 of the brownie batter into the prepared pan.
In a separate bowl, beat the cream cheese, 1/4 cup sugar, 1 egg, and 1/2 teaspoon vanilla until smooth.
Spread the cheesecake mixture over the brownie batter in the pan.
Dollop the remaining brownie batter and blackberry preserves over the top. Swirl with a knife to create a marbled effect.
Bake for 35-40 minutes, or until a toothpick inserted into the center comes out with a few moist crumbs.
Let cool completely before slicing and serving.