Question

D^2f, f=3x^4y-x^3y^2-x^5

Answer 1

Ответ:

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Answer 2

To find the second partial derivative of the function $f(x, y) = 3x^4y - x^3y^2 - x^5$, with respect to both $x$ and $y$, you will need to compute two partial derivatives.

First, find the partial derivative of $f$ with respect to $x$ ($∂f/∂x$):

$∂f/∂x = 12x^3y - 3x^2y^2 - 5x^4$

Now, find the partial derivative of $∂f/∂x$ with respect to $x$ again to get the second partial derivative $∂^2f/∂x^2$:

$∂^2f/∂x^2 = 36x^2y - 6xy^2 - 20x^3$

Next, find the partial derivative of $f$ with respect to $y$ ($∂f/∂y$):

$∂f/∂y = 3x^4 - 2x^3y$

Now, find the partial derivative of $∂f/∂y$ with respect to $y$ again to get the second partial derivative $∂^2f/∂y^2$:

$∂^2f/∂y^2 = -2x^3$

So, the second partial derivative of $f(x, y) = 3x^4y - x^3y^2 - x^5$ with respect to both $x$ and $y$ is:

$∂^2f/∂x^2 = 36x^2y - 6xy^2 - 20x^3$

$∂^2f/∂y^2 = -2x^3$