i know what duration is and how it relates to prices and interest rates, BUT can someone take the time to show me how duration is the 1st derivative of price-yield relationship, in other words, can someone prove to me mathematically and explain each step in the proof? I’m a bit fuzzy in calculus. thank you.

Not easy to do in a Forum…If i remember correctly the CFAI books lays it out in L1 books or mabye it was L2. Try Wikipedia or somethign similar.

Duration=Change of Price/Change of Interest Rate. Back to middle schoo, that is Rise/Run. Given the shape of Price of a bond vs interest rate is a curve, X square (like Bigwilly said, notation not easily written) must be involved in the equation. Take derivative of X square, it will yield a X term. Given Rise/Run, that is a 1st-degree equation. Let me make things more interesting for you, Convexity is the 2nd Derivative. Some parallel similiarity for option: Delta is the 1st Deriviative. Gamma is the 2nd Derivative. Have fun!!