Expert answers to "Find the total differential. z=xcos⁡y−ycos⁡x"

meplasemamiuk

Answered question

2021-12-03

Find the total differential.

z = x \cos y - y \cos x

mylouscrapza

Beginner2021-12-04Added 22 answers

The total differential of z=f(x,y) is:

d z = \frac{\partial z}{\partial x} d x + \frac{\partial z}{\partial y} d y

= f_{x} (x, y) d x + f_{y} (x, y) d y

To determine

f_{x} (x, y)

, take the derivative of the function with respect to x and treat y as constant.

f_{x} (x, y) = \frac{\partial}{\partial x} (x \cos y - y \cos x)

Apply the Sum/Difference Rule:

\frac{\partial}{\partial x} (f \pm g) = \frac{\partial f}{\partial x} \pm \frac{\partial g}{\partial x}

f_{x} (x, y) = \frac{\partial}{\partial x} (x \cos y - y \cos x)

= \frac{\partial}{\partial x} (x \cos (y)) - \frac{\partial}{\partial x} (y \cos (x))

= \cos (y) - (- y \sin (x))

= \cos (y) + y \sin (x)

To determine

f_{y} (x, y)

, take the derivative of the function with respect to y and treat x as constant.

f_{y} (x, y) = \frac{\partial}{\partial y} (x \cos y - y \cos x)

Apply the Sum/Difference Rule:

\frac{\partial}{\partial y} (f \pm g) = \frac{\partial f}{\partial y} \pm \frac{\partial g}{\partial y}

f_{y} (x, y) = \frac{\partial}{\partial y} (x \cos y - y \cos x)

= \frac{\partial}{\partial y} (x \cos (y)) - \frac{\partial}{\partial y} (y \cos (x))

= - x \sin (y) - \cos (x)

Substitute

f_{x} (x, y) and f_{y} (x, y)

in the equation (1), to obtain:

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