Rates of change using partial differentiation

If Z = f ( u , v , w , … ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq3668.tif"/> and du dt , dv dt , dw dt , … … https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq3669.tif"/> denote the rate of change of u, v, w, ….respectively, then the rate of change of Z, dZ dt https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq3670.tif"/> , is given by: (1) d Z d t = ∂ Z ∂ u d u d t + ∂ Z ∂ v d v d t + ∂ Z ∂ w d w d t + .. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429294402/968186b9-ffa4-4f2b-aa54-7fc58a604561/content/eq3671.tif"/>