Question

Determine the partial derivatives of \(\frac{{\partial z}}{{\partial x}}\) and \(\frac{{\partial z}}{{\partial y}}\) using equation 7.

\(xyz = \cos (x + y + z)\)

Short answer

The partial derivatives \(\frac{{\partial z}}{{\partial x}}\) and \(\frac{{\partial z}}{{\partial y}}\) with the help of equation 7 are \(\frac{{yz + \sin (x + y + z)}}{{xy + \sin (x + y + z)}}\) and\(\frac{{xz + \sin (x + y + z)}}{{xy + \sin (x + y + z)}}\), respectively.

Step-by-step solution

Formula used

Equation 7: "\(\frac{{\partial z}}{{\partial x}} = - \frac{{\frac{{\partial F}}{{dx}}}}{{\frac{{\partial F}}{{dz}}}}\)and\(\frac{{\partial z}}{{\partial y}} = - \frac{{\frac{{\partial F}}{{\partial y}}}}{{\frac{{\partial F}}{{\partial z}}}}\)

The function\(F\)becomes\(F(x) = xyz - \cos (x + y + z)\).

Determine the partial derivatives of \(\frac{{\partial z}}{{\partial x}}\)and\(\frac{{\partial z}}{{\partial y}}\)

As given equation is \(xyz = \cos (x + y + z)\)

That is, \(xyz - \cos (x + y + z) = 0\)

Obtain the value of \(\frac{{\partial z}}{{\partial x}}\) as follows,

\(\begin{aligned}{l}\frac{{\partial z}}{{\partial x}} = - \frac{{\frac{\partial }{{dx}}(xyz - \cos (x + y + z))}}{{\frac{\partial }{{dz}}(xyz - \cos (x + y + z))}}\\\frac{{\partial z}}{{\partial x}} = \frac{{yz - ( - \sin (x + y + z))}}{{xy - ( - \sin (x + y + z))}}\\\frac{{\partial z}}{{\partial x}} = \frac{{yz + \sin (x + y + z)}}{{xy + \sin (x + y + z)}}\end{aligned}\)

Obtain the value of \(\frac{{\partial z}}{{\partial y}}\) as follows,

\(\begin{aligned}{l}\frac{{\partial z}}{{\partial y}} = - \frac{{\frac{\partial }{{\partial y}}(xyz - \cos (x + y + z))}}{{\frac{\partial }{{\partial z}}(xyz - \cos (x + y + z))}}\\\frac{{\partial z}}{{\partial y}} = \frac{{xz - ( - \sin (x + y + z))}}{{xy - ( - \sin (x + y + z))}}\\\frac{{\partial z}}{{\partial y}} = \frac{{xz + \sin (x + y + z)}}{{xy + \sin (x + y + z)}}\end{aligned}\)

Therefore, the partial derivatives \(\frac{{\partial z}}{{\partial x}}\) and \(\frac{{\partial z}}{{\partial y}}\) with the help of equation 7 are \(\frac{{yz + \sin (x + y + z)}}{{xy + \sin (x + y + z)}}\) and\(\frac{{xz + \sin (x + y + z)}}{{xy + \sin (x + y + z)}}\), respectively.