Derivatives of Polynomials and Exponential Functions

Delta for a polynomial payoff, the sensitivity of a discount factor $e^{-rT}$ to changes in the rate, the acceleration of a particle given its position — all require differentiating polynomials and exponential functions. Computing these from the limit definition each time is laborious. This note derives three rules — the constant rule, the power rule, and the derivative of $e^x$ — that together let you differentiate any polynomial or exponential expression in one line. The fact that $e^x$ is its own derivative is one of the most consequential properties in all of mathematics, and this note shows why $e$ is the only base for which it holds.

• differentiate any constant, and justify the derivative of a constant as zero;
• state and apply the power rule for any real exponent $n$;
• apply the constant-multiple, sum, and difference rules to differentiate any polynomial;
• state and apply $\tfrac{d}{dx}e^x = e^x$;
• find horizontal tangents and evaluate derivatives of motion equations.

1. Derivative of a constant
2. The power rule
3. Constant-multiple, sum, and difference rules
4. Derivative of the natural exponential function
5. Higher derivatives of polynomials