Answer to Question #210052 in Calculus for Faith

Question #210052

Determine the first order partial derivative of the following functions:

1.z=in(x+t²).

2.f(x,y,z)=xy²e^-xz.


1
Expert's answer
2021-06-28T03:31:31-0400

1.


z=ln(x+t2)z=\ln(x+t^2)


zx=zx=1x+t2z_x=\dfrac{\partial z}{\partial x}=\dfrac{1}{x+t^2}

zt=zt=2tx+t2z_t=\dfrac{\partial z}{\partial t}=\dfrac{2t}{x+t^2}



2.


f(x,y,z)=xy2exzf(x,y,z)=xy^2e^{-xz}

fx=fx=y2exzxy2zexzf_x=\dfrac{\partial f}{\partial x}=y^2e^{-xz}-xy^2ze^{-xz}

fy=fy=2xyexzf_y=\dfrac{\partial f}{\partial y}=2xye^{-xz}

fz=fx=x2y2exzf_z=\dfrac{\partial f}{\partial x}=-x^2y^2e^{-xz}




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