*5*

*Partial differentiation*

**5.1***Using the appropriate properties of ordinary derivatives, perform the following.*

(a) *Find all the first partial derivatives of the following functions f*(*x*, *y*)*:*

(i) *x*^{2}*y,* (ii) *x*^{2} + *y*^{2} + 4*,* (iii) sin(*x/y*)*,* (iv) tan^{–1}(*y/x*)*,*

(v) *r*(*x, y, z*) = (*x*^{2} + *y*^{2} + *z*^{2})^{1/2}.

(b) *For* (i)*,* (ii) *and* (v)*, find ∂*^{2}*f/∂x*^{2}*,* *∂*^{2}*f/∂y*^{2} *and ∂*^{2}*f/∂x∂y.*

(c) *For* (iv) *verify that ∂*^{2}*f/∂x∂y* = *∂*^{2}*f/∂y∂x.*

These are all straightforward applications of the ...

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