The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t

The natural logarithm as the inverse of the exponential function

Derivative of an inverse: if $f$ is continuous and injective on a nondegenerate interval $I$ and differentiable at $c\in I$ with $f^{\mathrm{\prime }}(c)\mathrm{\ne }0$, then the inverse $g$ is differentiable at $f(c)$ with $g^{\mathrm{\prime }}(f(c))=1\mathrm{/}{f}^{\mathrm{\prime }}(c)$; and if $f^{\mathrm{\prime }}(c)=0$ then $g$ is not differentiable at $f(c)$

The exponential function is smooth and $(\exp {)}^{\mathrm{\prime }}=\exp$

The first fundamental theorem: if $f$ is integrable on $[a,b]$ and continuous at $c$, then $F^{\mathrm{\prime }}(c)=f(c)$; in particular a continuous $f$ has $F$ as a primitive

The integral with oriented limits: ${\int }_a^{a}f:=0$ and ${\int }_b^{a}f:=-{\int }_a^{b}f$

Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function

A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$; consequently two such functions with the same derivative differ by a constant