Question

Find both first partial derivatives. z = In (x/y).

Answer 1

The partial derivatives are,

w.r.t x

`\partial z/ \partial x = 1/x`

And , w.r.t y

`\partial z/ \partial y= -1/y`

Explanation:

z = In (x/y).

Calculating both partial derivatives (with respect to x and y)

Wrt x,

wrt x, we get,

`z = In (x/y).\\\partial/ \partial x[z]=\partial/ \partial x[ln(x/y)]\\\partial z/ \partial x = 1/(x/y)(\partial/ \partial x[x/y])\\\partial z/ \partial x = y/(x)(1/y)\\\partial z/ \partial x = 1/x`

Now,

wrt y,

we get,

`z = In (x/y).\\\partial / \partial y[z]=\partial / \partial y[ln(x/y)]\\\partial z/ \partial y =(1/(x/y)) \partial/ \partial y [x/y]\\\partial z/ \partial y = y/x(-1)(x)(1/y^2)\\\partial z/ \partial y= -1/y`

So, we have found both first partial derivatives.