Question #339770

Suppose z is a function of x and y, and tany2+x2 = zxe6y. Solve for ∂z/∂x and ∂z/∂y.


1
Expert's answer
2022-05-13T05:19:06-0400
y2+x2=zxe6y\sqrt{y^2+x^2}=z^xe^{6y}

12ln(y2+x2)=xlnz+6y\dfrac{1}{2}\ln(y^2+x^2)=x\ln z+6y

Differentiate both sides with respect to xx


2x2(y2+x2)=lnz+xz(zx)\dfrac{2x}{2(y^2+x^2)}=\ln z+\dfrac{x}{z}(\dfrac{\partial z}{\partial x})


zx=zy2+x2zlnzx\dfrac{\partial z}{\partial x}=\dfrac{z}{y^2+x^2}-\dfrac{z\ln z}{x}



12ln(y2+x2)=xlnz+6y\dfrac{1}{2}\ln(y^2+x^2)=x\ln z+6y

Differentiate both sides with respect to yy


2y2(y2+x2)=xz(zy)+6\dfrac{2y}{2(y^2+x^2)}=\dfrac{x}{z}(\dfrac{\partial z}{\partial y})+6

zy=yzx(y2+x2)6zx\dfrac{\partial z}{\partial y}=\dfrac{yz}{x(y^2+x^2)}-\dfrac{6z}{x}


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