The Product Rule, Quotient Rule, and Derivatives of Trigonometric Functions

A bond's dollar duration is the product of its modified duration and its price — differentiating this product requires the product rule. Option Greeks for more complex payoffs frequently involve quotients of continuous functions. And the pricing of periodic structures (coupon bonds, swap payments) involves derivatives of trigonometric functions of time. This note derives the product and quotient rules, proves that $(d/dx)\sin x = \cos x$ (using the addition formula for sine and the key limit $\lim_{\theta\to 0}\sin\theta/\theta = 1$), and collects the six trig derivatives into one table.

• state and apply the product rule in both prime and Leibniz form;
• state and apply the quotient rule and identify the common sign error;
• state and verify from the definition that $(\sin x)' = \cos x$ and $(\cos x)' = -\sin x$;
• derive and apply the four remaining trig derivatives using the quotient rule;
• differentiate any expression built from polynomials, exponentials, and trig functions.

1. The product rule
2. The quotient rule
3. The key trig limits
4. Derivative of $\sin x$
5. The six trigonometric derivatives