These are gorgeous coins. The one in the picture is superb, with full mint luster.
A US $10 gold piece contains 0.4838 troy ounces of pure gold. Multiply that by the current price of gold (e.g., from kitco.com), and you get the base value for the coin. For $1295 per ounce of gold, for instance, the base value is 0.4838 x 1295 = $626.50. Check the value of gold right away. It changes constantly!
To the base value add $100 to $200 for numismatic (coin collector) premium, $100 for average circulated, $200 for fully uncirculated. This is how much collectors are normally willing to pay over and above the base value. Fully uncirculated coins in numismatic slabs are priced with numismatic premiums over $200.
If you were to sell your coin to a dealer, he or she would probably pay 25% to 50% less, with the margin allowing the dealer to stay in business. Selling it to a collector directly would bring the retail amount.
Just about every date in this series follows the price rules outlined above (that is, base gold value plus collector premium). However, there are some really special dates in this series of coins. Certain specimens dated 1907 are worth over $50000 when small periods appear near E PLURIBUS UNUM. The 1920S and 1930S dates and mint marks also catalog over $10000 in worn condition. Look for the mint mark on the reverse, under the eagle on the left. The 1911D is a more valuable than most, due to its low mintage.
See this page for more info on counterfeits of this coin.