Question

If one quantity Z is a function of two other quantities X and Y , Z = Z(X, Y ), then their derivatives with respect to each other are related as ∂X ∂Y Z ∂Y ∂Z X ∂Z ∂X Y = −1.

Answer 1

It is given that, One Quantity Z is a function of two other quantities X and Y such that

Z=Z(x,y)

Taking Partial Derivative of Z with respect to x and y ,and then partial derivative of x with respect to y,and then Multiplying the three equations, we get

$\rightarrow1=y\times \frac{\partial x}{\partial z}----(1)\\\\\rightarrow1=x\times \frac{\partial y}{\partial z}----(2)\\\\\rightarrow x\times \frac{\partial y}{\partial x}+y=0\\\\\rightarrow \frac{\partial y}{\partial x}=\frac{-y}{x}------(3)\\\\1 \times 2 \times 3\\\\\rightarrow \frac{\partial x}{\partial y} \times \frac{\partial y}{\partial z} \times \frac{\partial z}{\partial x}=\frac{-x}{y} \times \frac{1}{x} \times y\\\\= -1$

Cancelling terms from numerator and Denominator to Obtain the result.