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MATH 2204 Lecture Notes - Lecture 9: Partial Derivative

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14 Oct 2018

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Daniel eisert | math-2204: definition of the partial derivative of f with respect to x at (a, b): This definition holds true similarly with the partial derivative of f with respect to y at (a, b). (cid:3051)(cid:4666)(cid:1853),(cid:1854)(cid:4667)=lim (cid:2868)(cid:4666)(cid:1853)+ ,(cid:1854)(cid:4667) (cid:4666)(cid:1853),(cid:1854)(cid:4667) (cid:3052)(cid:4666)(cid:1853),(cid:1854)(cid:4667)=lim (cid:2868)(cid:4666)(cid:1853)+ ,(cid:1854)(cid:4667) (cid:4666)(cid:1853),(cid:1854)(cid:4667) These notations can be applied similarly with y: notation: if (cid:1878)=(cid:4666)(cid:1876),(cid:1877)(cid:4667), we write: (cid:3051)(cid:4666)(cid:1876),(cid:1877)(cid:4667)=(cid:3051)=(cid:1876)=(cid:1876)(cid:4666)(cid:1876),(cid:1877)(cid:4667)=(cid:1878)(cid:1876)=(cid:2869)=(cid:2869)=(cid:3051) Rules for finding partial derivatives of (cid:1878)=(cid:4666)(cid:1876),(cid:1877)(cid:4667): to find (cid:3051) regard y as a constant and differentiate (cid:4666)(cid:1876),(cid:1877)(cid:4667) with respect to x, to find (cid:3052) regard x as a constant and differentiate (cid:4666)(cid:1876),(cid:1877)(cid:4667) with respect to y. Interpretation of partial derivatives: partial derivatives (cid:3051)(cid:4666)(cid:1853),(cid:1854)(cid:4667) and (cid:3052)(cid:4666)(cid:1853),(cid:1854)(cid:4667) can be interpreted geometrically as the slopes of the tangent lines at (cid:4666)(cid:1853),(cid:1854),(cid:1855)(cid:4667) to the traces c1 and c2 of s in the planes (cid:1877)=(cid:1854) and (cid:1876)=(cid:1853). They can also be interpreted as rates of change. If (cid:1878)=(cid:4666)(cid:1876),(cid:1877)(cid:4667), then (cid:3053)(cid:3051) represents the rate of change of z with respect to x when y is fixed.

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