Ifu =e^(x//y)+e^(y//z)+e^(z//x) then show that x*(del u)/(del x)+y*(del u)/(del y)+z(del u)/(del z)=0

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Ifu  =e^(x//y)+e^(y//z)+e^(z//x) then show that  x*(del u)/(del x)+y*(del u)/(del y)+z(del u)/(del z)=0

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Find Partial Derivative of u with x: Let's start by finding the partial derivative of u with respect to x. u = e^(x/y) + e^(y/z) + e^(z/x) (del u)/(del x) = (del)/(del x)(e^(x/y)) + (del)/(del x)(e^(y/z)) + (del)/(del x)(e^(z/x)) Since y and z are treated as constants when taking the partial derivative with respect to x, the second and third terms become zero. (del u)/(del x) = (del)/(del x)(e^(x/y)) + 0 + 0 Now, let's find the derivative of e^(x/y) with respect to x. (del)/(del x)(e^(x/y)) = (1/y)e^(x/y) So, (del u)/(del x) = (1/y)e^(x/y)Find Partial Derivative of u with y: Next, we find the partial derivative of u with respect to y. (del u)/(del y) = (del)/(del y)(e^(x/y)) + (del)/(del y)(e^(y/z)) + (del)/(del y)(e^(z/x)) The first term involves the chain rule, and the second term is a direct derivative, while the third term becomes zero because x and z are treated as constants. (del u)/(del y) = -(x/y^2)e^(x/y) + (1/z)e^(y/z) + 0Find Partial Derivative of u with z: Now, we find the partial derivative of u with respect to z. (del u)/(del z) = (del)/(del z)(e^(x/y)) + (del)/(del z)(e^(y/z)) + (del)/(del z)(e^(z/x)) The first and second terms become zero because x and y are treated as constants, and the third term involves the chain rule. (del u)/(del z) = 0 + 0 - (y/z^2)e^(y/z)Multiply and Sum Partial Derivatives: We now multiply each partial derivative by its corresponding variable and sum them up. x*(del u)/(del x) + y*(del u)/(del y) + z*(del u)/(del z) = x*(1/y)e^(x/y) - y*(x/y^2)e^(x/y) + y*(1/z)e^(y/z) - z*(y/z^2)e^(y/z) Simplify the expression by combining like terms. = (x/y)e^(x/y) - (x/y)e^(x/y) + (y/z)e^(y/z) - (y/z)e^(y/z) = 0

If $u = e^{\frac{x}{y}} + e^{\frac{y}{z}} + e^{\frac{z}{x}}$ then show that $x\left(\frac{\partial u}{\partial x}\right) + y\left(\frac{\partial u}{\partial y}\right) + z\left(\frac{\partial u}{\partial z}\right) = 0$

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