| function | differential law |
|---|---|
| constant | \( {\displaystyle \diff (c) = 0 } \) |
| constant-adder | \( {\displaystyle \diff (u + c) = \diff u } \) |
| constant-multiplier | \( {\displaystyle \diff (c \mult u) = c \mult \diff u } \) |
| addition | \( {\displaystyle \diff (u + v) = \diff u + \diff v } \) |
| negation | \( {\displaystyle \diff (- v) = - \diff v } \) |
| subtraction | \( {\displaystyle \diff (u - v) = \diff u - \diff v } \) |
| multiplication | \( {\displaystyle \diff (u \mult v) = v \mult \diff u + u \mult \diff v } \) |
| reciprocal | \( {\displaystyle \diff \biggl(\frac{1}{v}\biggr) = \frac{- 1}{v^2} \mult \diff v } \) |
| division | \( {\displaystyle \diff \biggl(\frac{u}{v}\biggr) = \frac{1}{v} \mult \diff u - \frac{u}{v^2} \mult \diff v } \) |
| squaring | \( {\displaystyle \diff \bigl(u^2\bigr) = 2 u \mult \diff u } \) |
| cubing | \( {\displaystyle \diff \bigl(u^3\bigr) = 3 u^2 \mult \diff u } \) |
| square root | \( {\displaystyle \diff \bigl(\sqrt[2]{u}\bigr) = \frac{1}{2 \mult \sqrt[2]{u}} \mult \diff u } \) |
| cube root | \( {\displaystyle \diff \bigl(\sqrt[3]{u}\bigr) = \frac{1}{3 \mult \bigl(\sqrt[3]{u}\bigr)^2} \mult \diff u } \) |
| base | \( {\displaystyle \diff \bigl(v^c\bigr) = c \mult v^{c - 1} \mult \diff v } \) |
| exponential | \( {\displaystyle \diff \bigl(c^u\bigr) = c^u \mult \log c \mult \diff u } \) |
| power | \( {\displaystyle \diff \bigl(v^u\bigr) = v^u \mult \log v \mult \diff u + u \mult v^{u-1} \mult \diff v } \) |
| natural exponential | \( {\displaystyle \diff (\exp u) = \exp u \mult \diff u } \) |
| natural logarithm | \( {\displaystyle \diff (\log u) = \frac{1}{u} \mult \diff u } \) |
| cosine | \( {\displaystyle \diff (\cos u) = - \sin u \mult \diff u } \) |
| sine | \( {\displaystyle \diff (\sin u) = \cos u \mult \diff u } \) |