As you approach the water here, the spider will re-appear behind you. It will chase you now, so you will need to pick up the pace a bit because it will destroy you if it catches up with you. Push the log into the water, then jump onto it, and then across to the other side.
You will come to a see-saw like log that you will need to use to get up to the higher platform on the right. Stand on the far right side of the stump and wait for the spider to stand on the left hand side. As it does, quickly jump across to the next platform.
Continue up the hill and push the stump over. Use it to jump across the gap, climb up the ledge here and continue up the hill until you reach the rock held in place by a stick. Jump on the stick, then run back down the hill and hug the wall closest to the ledge. The rock should roll over your head and take out the spider. Continue onwards until you reach a spike pit. Two shadowy figures on the opposite side will see you and run away.
Turn around and you will see the spider is back again. Move towards it until it performs a stabbing attack, avoid this and then grab the leg and pull it. It should come off in your hands. Now that it is pretty much harmless, simply jump over the spider's body and then grab on and push the arachnid's body into the spike pit to the right.
Use the body to jump across to the other side.