slip (v.)
early 14c., "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 (cognates: Old High German slifan, Middle Dutch slippen, German schleifen "to glide, slide"), from PIE *sleib-, from root *(s)lei- "slimy, sticky, slippery" (see slime (n.)).

From mid-14c. with senses "lose one's footing," "slide out of place," "fall into error or fault." Sense of "pass unguarded or untaken" is from mid-15c. That of "slide, glide" is from 1520s. Transitive sense from 1510s; meaning "insert surreptitiously" is from 1680s. Related: Slipped; slipping. To slip up "make a mistake" is from 1855; to slip through the net "evade detection" is from 1902.
slip (n.2)
in various senses from slip (v.). Meaning "act of slipping" is from 1590s. Meaning "mistake, minor fault, blunder" is from 1610s. Sense of "woman's sleeveless garment" (1761) is from notion of something easily slipped on or off (compare sleeve). To give (someone) the slip "escape from" is from 1560s. Meaning "landing place for ships" is mid-15c.; more technical sense in ship-building is from 1769. Slip of the tongue (1725) is from earlier slip of the pen (1650s), which makes more sense as an image.
slip (n.1)
mid-15c., "edge of a garment;" 1550s, "narrow strip," probably from Middle Low German or Middle Dutch slippe "cut, slit," possibly related to Old English toslifan "to split, cleave." Sense of "narrow piece of paper" (as in pink slip) in 1680s.
slip (n.3)
"potter's clay," mid-15c., "mud, slime," from Old English slypa, slyppe "slime, paste, pulp, soft semi-liquid mass," related to slupan "to slip" (see sleeve).
slip (n.4)
"sprig or twig for planting or grafting, small shoot," late 15c., of uncertain origin. Compare Middle Dutch slippe, German schlippe, schlipfe "cut, slit, strip." Hence "young person of small build" (1580s, as in a slip of a girl); see slip (n.1).