27 higher derivatives for functions of two variables

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2.7Higher Derivatives for Functions of Two Variables1.If z=f(x , y)is a function of two variables, then fx(x , y)and fy(x , y)also functions of two variables. Partial derivatives of fx(x , y)and fy(x , yare the second partial derivatives.2.They are denoted by the following notations.fxx(x, y)=fxx=(fx)x=∂x(∂f∂x)=2f∂ x2=2∂ x2f(x , y)=2z∂x2=f11fxy(x, y)=fxy=(fx)y=∂ y(∂ f∂x)=2f∂ y ∂ x=2∂ y ∂xf(x, y)=2z∂ y ∂x=f12are)
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.2.7Partial Derivatives for Functions of More Than Two Variables1.Partial derivatives can also be defined for functions of three or morevariables.
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Definition 6Suppose that fxand fyare defined throughout an open region Dcontainingthe point (a,b)and that fxand fyare continuous at (a,b). If z=f(x , y),then fis differentiable at (a,b)if fx(a,b)and fy(a,b)exist and ∆ zsatisfies an equation of the form∆ z=fx(a,b)∆ x+fy(a,b)∆ y+ε1∆ x+ε2∆ ywhere ε1and ε20as ∆ xand ∆ y →0. We call fdifferentiable if it isdifferentiable at every point in its domain, and say that its graph is a smoothsurface.1.From Definition 6:a)∆ xis an increment in x; represents the change in the value ofxwhen xchanges from ato a+∆ x.b)∆ yis an increment in y; represents the change in the value ofywhen ychanges from bto b+∆ y.

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