Find all the second partial derivatives v=xyx−y

Question

Find all the second partial derivatives  v=xyx−y

Nichole Watt · Accepted Answer

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alenahelenash · Accepted Answer

Answer:∂2v∂x2=−2y2(x−y)3∂2v∂y2=2x2(x−y)3∂2v∂y∂x=−2y(x−y)2+4y2(x−y)3∂2v∂x∂y=−2x(x−y)2+4x2(x−y)3Explanation:First, we differentiate v with respect to x, treating y as a constant:∂v∂x=y(x−y)−xy(x−y)2=y2(x−y)2Next, we differentiate v with respect to y, treating x as a constant:∂v∂y=x(x−y)−xy(x−y)2=−x2(x−y)2Now, let&#039;s differentiate each of these partial derivatives with respect to the respective variable.Taking the second partial derivative with respect to x, we differentiate ∂v∂x with respect to x:∂2v∂x2=∂∂x(y2(x−y)2)=−2y2(x−y)3Next, we take the second partial derivative with respect to y, differentiating ∂v∂y with respect to y:∂2v∂y2=∂∂y(−x2(x−y)2)=2x2(x−y)3Finally, we find the mixed partial derivatives by differentiating the first partial derivative with respect to the other variable.Taking the second partial derivative with respect to y of ∂v∂x:∂2v∂y∂x=∂∂y(y2(x−y)2)=−2y(x−y)2+4y2(x−y)3And taking the second partial derivative with respect to x of ∂v∂y:∂2v∂x∂y=∂∂x(−x2(x−y)2)=−2x(x−y)2+4x2(x−y)3

karton · Accepted Answer

Step 1. The partial derivative of v with respect to x:∂v∂x=y(x−y)−xy(x−y)2Step 2. The partial derivative of ∂v∂x with respect to x (second partial derivative with respect to x):∂2v∂x2=2y(x−y)−2xy(x−y)3Step 3. The partial derivative of v with respect to y:∂v∂y=x(x−y)−xy(x−y)2Step 4. The partial derivative of ∂v∂y with respect to y (second partial derivative with respect to y):∂2v∂y2=2x(x−y)−2xy(x−y)3Step 5. The mixed partial derivative of v with respect to x and y (second partial derivative with respect to x and y):∂2v∂x∂y=y(x−y)−x(x−y)(x−y)3Step 6. The mixed partial derivative of v with respect to y and x (second partial derivative with respect to y and x):∂2v∂y∂x=y(x−y)−x(x−y)(x−y)3

star233 · Accepted Answer

To find all the second partial derivatives of the function

v = \frac{x y}{x - y}

, we need to differentiate it twice with respect to both

x

and

y

. Let's proceed with the calculations.
First, we find the partial derivative of

v

with respect to

x

:

\frac{\partial v}{\partial x} = \frac{y (x - y) - x y}{(x - y)^{2}} = \frac{y^{2}}{(x - y)^{2}}

Next, we differentiate

\frac{\partial v}{\partial x}

with respect to

x

again:

\frac{\partial^{2} v}{\partial x^{2}} = \frac{2 y^{2} (x - y) - 2 y (x - y) y}{(x - y)^{4}} = \frac{2 y^{2} (x - 2 y)}{(x - y)^{3}}

Now, we find the partial derivative of

v

with respect to

y

:

\frac{\partial v}{\partial y} = \frac{x (x - y) - x y (- 1)}{(x - y)^{2}} = \frac{x^{2}}{(x - y)^{2}}

Finally, we differentiate

\frac{\partial v}{\partial y}

with respect to

y

again:

\frac{\partial^{2} v}{\partial y^{2}} = \frac{2 x^{2} (x - y) - 2 x (x - y) x}{(x - y)^{4}} = \frac{2 x^{2} (y - 2 x)}{(x - y)^{3}}

Hence, the second partial derivatives of

v

are:

\frac{\partial^{2} v}{\partial x^{2}} = \frac{2 y^{2} (x - 2 y)}{(x - y)^{3}}

and

\frac{\partial^{2} v}{\partial y^{2}} = \frac{2 x^{2} (y - 2 x)}{(x - y)^{3}}

These are the solutions for the second partial derivatives of

v

.

Find all the second partial derivatives v=\frac{xy}{x-y}

Answered question

Answer & Explanation

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