Question

(Assume that 9 is the symbol for a partial derivative) For the equation (1/x)+(1/y)+(1/z)=1, evaluate 9z/9x and 9z/9y at the point (2,3,6)

Answer 1

given expression is,

$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1$

differentiate with respect to x

$\frac{-1}{x^2}-\frac{1}{z^2}\frac{\delta z}{\delta x} =0$

$\frac{\delta z}{\delta x}=-\frac{z^2}{x^2}$

$\frac{\delta z}{\delta x}_{(2,3,6)}=-\frac{6^2}{2^2}$

=$-\frac{36}{4}=-9$

Now partially differentiate given expression with respect to y:

$\to$ $\frac{-1}{y^2}-\frac{1}{z^2}\frac{\delta z}{\delta y} =0$

$\to \frac{\delta z}{\delta x} =\frac{-z^2}{y^2}$

$\to \frac{\delta z}{\delta x}_{(2,3,6)}=-\frac{6^2}{3^2}$

=$-\frac{36}{9}$

=-4