slip v.

early 14c., slippen, "to escape, to move softly and quickly," from an unrecorded Old English word or cognate Middle Low German slippen "to glide, slide," from Proto-Germanic *slipan (source also of Old High German slifan, Middle Dutch slippen, German schleifen "to glide, slide"). This is probably from PIE *sleib-"slip, slide," from root *(s)lei- "slimy, sticky, slippery" (see slime (n.)). The verb is not found in Old English, which did have related adjective slipor "slippery, having a smooth surface." Related: Slipped; slipping. 

It is attested from mid-14c. in the sense of "lose one's footing, slide suddenly and unawares," also "slide out of place," also "fall into error or fault." The meaning "pass unguarded or untaken" is from mid-15c. That of "slide, glide, pass smoothly and easily" is from 1520s.

The transitive sense of "cause to move with a sliding motion" is from 1510s; the meaning "insert surreptitiously, put or place secretly" is from 1680s. The meaning "let loose, release from restraint" (1580s), is probably from the noun sense of "leash for a (hunting) dog that can be easily released" (1570s).

To slip on "put on (clothing, etc.) loosely or in haste" is from 1580s; to slip off "take off noiselessly or hastily" is from 1590s. To slip up "make a mistake, err inadvertently" is from 1855; to slip through the net "evade detection" is by 1829 (for slip through the cracks see crack (n.)). To let (something) slip originally (1520s) was a reference to hounds on a leash; figurative use for "allow to escape through carelessness" is by 1540s.

"long, narrow, and more or less rectangular piece," mid-15c., originally "edge of a garment;" by 1550s generally as "narrow strip;" probably from Middle Low German or Middle Dutch slippe "cut, slit," which might be related to Old English toslifan "to split, cleave." The sense of "narrow piece of paper" (as in pink slip) by 1680s. From the same source as slip (n.4).

in numerous and various senses from slip (v.), late 13c. as a surname. The meaning "sloping landing place for ships between wharves or a dock" is by mid-15c. The meaning "act of slipping" (as walking on ice) is from 1590s. The meaning "mistake, fault, blunder," especially if minor or unintended is from 1570s.

The sense of "woman's sleeveless garment" (1761) is from notion of something easily "slipped" on or off (compare sleeve). Originally an outer garment; in 20c. of a sleeveless undergarment or petticoat.

To give (someone) the slip "escape from" is from 1560s. Slip of the tongue is 1725 (from Latin lapsus linguae); slip of the pen (Latin lapsus calami) is 1650s.

mid-15c., "mud, slime; curdled milk," from Old English slypa, slyppe "slime, paste, pulp, soft semi-liquid mass," which is related to slupan "to slip" (from PIE root *sleubh- "to slide, slip," source of slop). The meaning "potters' clay reduced to a semi-fluid condition" is by 1630s and through it the old word is preserved in technical terms.

"sprig or twig detached from a main stock for planting or grafting, small shoot," late 15c., a word of uncertain origin. Middle Dutch slippe, German schlippe, schlipfe "cut, slit, strip of wood" have been compared, but the chronology of the sense evolution seems not to match. Compare slip (n.1). Hence "young person of small build" (1580s, as in slip of a girl).