Delta is the rate at which an option price changes per unit change in the underlying. Theta is the rate of price change per unit of time. Both are derivatives in the calculus sense — limits of difference quotients. This note defines the derivative, establishes that it is simultaneously the slope of a tangent line and an instantaneous rate of change, extends the single-point definition to the derivative function , and identifies exactly when a function fails to be differentiable. The core relationship — differentiability implies continuity, but not vice versa — governs which functions admit a well-defined Greek and which do not.
Learning objectives. By the end of this note you should be able to:
- write the two equivalent forms of the derivative definition (via and ) and evaluate either form for polynomial and simple rational functions;
- interpret as the slope of the tangent line and as the instantaneous rate of change of at ;
- compute as a function by applying the limit definition to a general ;
- state and prove that differentiability implies continuity, and give a counterexample to the converse;
- identify the three geometric failure modes for differentiability: corners, discontinuities, and vertical tangents.
In this lesson
- The tangent line as a limit
- The derivative at a point
- Instantaneous rate of change
- The derivative as a function
- Higher derivatives
- Differentiability and continuity
- The three failure modes
Before you start
- Calculating Limits: Limit Laws and Algebraic Techniques
- Continuity, the Intermediate Value Theorem, and the Extreme Value Theorem
The full lesson, with worked examples and exercise solutions, is for members.
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