Question

Find the total differential. z = x cos(y) − y cos(x)

Answer 1

The total differential of the function z = x cos(y) − y cos(x) is found by taking the partial derivatives with respect to x and y, resulting in dz = (cos(y) + y sin(x))dx + (−x sin(y) − cos(x))dy.

Step-by-step explanation:

To find the total differential of the function z = x cos(y) − y cos(x), we need to calculate the partial derivatives of z with respect to x and y and then combine them into the differential equation.

The partial derivative of z with respect to x is:

∂z/∂x = cos(y) + y sin(x)

And the partial derivative of z with respect to y is:

∂z/∂y = −x sin(y) − cos(x)

Combining these into the total differential, we get:

dz = (∂z/∂x)dx + (∂z/∂y)dy

dz = (cos(y) + y sin(x))dx + (−x sin(y) − cos(x))dy

This equation represents the total differential of the given function.

Answer 2

`z=x\cos y-y\cos x`


`\mathrm dz=(\cos y\,\mathrm dx-x\sin y\,\mathrm dy)-(-y\sin x\,\mathrm dx+\cos x\,\mathrm dy)`

`\mathrm dz=(\cos y-y\sin x)\,\mathrm dx-(x\sin y+\cos x)\,\mathrm dy`