Find the first partial derivatives of f(x,y,z) = z arctan(y/x) at the point (4, 4, 5)

partial-derivatives

#1

Find the first partial derivatives of f(x,y,z) = z arctan(y/x) at the point (4, 4, 5).

Answer:

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F(x,y,z)=z*arctan(y/x)

d/dx arctan(x)=1/(1+(x^2))

differentiate applying product rule, assuming that y and z are constants

df/dx=z*(1/(1+(y/x)^2))*(-y/x^2)
=-(zy/(x^2(1+(y/x)^2)))

differentiate applying product rule, assuming that x and z are constants

df/dy=z*(1/(1+(y/x)^2))*(1/x)
=-(z/(x(1+(y/x)^2)))

differentiate applying product rule, assuming that x and y are constants

df/dz=1*arctan(y/x)
=arctan(y/x)

evaluate using each derivative:

dF(x,y,z)/dx=-5/8

dF(x,y,z)/dy=5/4

dF(x,y,z)/dz=-(pi/4)rad or 45°