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SQQM2033 Intermediate Calculus CHAPTER 2 PARTIAL DERIVATIVE2.1 Simple Partial DerivativeDefinition 1 PARTIAL DERIVATIVEIf z= f(x,y), the partial derivative of fwith respect to x, denoted fx, is the function given by:fx(x,y) = hyxfyhxfh),(),(lim0provided that the limit exists.If z= f(x,y), the partial derivative of fwith respect to y, denoted fy, is the function given by:fy(x,y) = hyxfhyxfh),(),(lim0provided that the limit exists.By analyzing the foregoing definition, we can state the following procedure to find fxandfy.Note: To compute the partial derivatives, just use ordinary derivative of a function single variable and not use by definition 2.1.1 and 2.1.2. Ex 1Find 1,3xfand 1,3yffor the function 32,224fx yx yyx.1Procedure to find fx(x,y) and fy(x,y)To find fx, treat yas a constant, and differentiate f withrespect to xin the usual way.To find fy, treat xas a constant, and differentiate f withrespect to yin the usual wayNotations for partial derivatives: If z= f(x,y), we writefx(x,y) = fx= 11( ,)xfzDf x yfffDxxxfy(x,y) = fy= 22( ,)yfzf x yfffDDyyy
SQQM2033 Intermediate Calculus Ex 2If f(x, y) = x3+ x2y3– 2y2find fx(2, 1) and fyTo give a geometric interpretation of partial derivatives, we recall that the equation z= f(x, y) represents a surface S (the graph of f). If f(a, b) = c, then the point P(a, bon SEx3Let 23,5fx yx yy(a)Find the slope of the surface z= f(x, y) in the x-direction at the point (1, – 2).(b) Find the slope of the surface z= f(x, y) in the y-direction at the point (1, – 2). (2, 1), c) lies..
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SQQM2033 Intermediate Calculus 2.2 HIGHER ORDER PARTIAL DERIVATIVES
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