Question

u=x^y
show del^3u/delx^2dely = del^3u/del x del y del x​

Answer 1

$del^3u/delx^2dely = del^3u/ \del x \del y \del x$

To show that del^3u/delx^2dely = del^3u/del x del y del x, we can use the properties of partial derivatives.

The partial derivative of a function with respect to a variable is the rate of change of that function with respect to that variable while keeping the other variables constant.

The partial derivative of u with respect to x is:

$delu/delx = yx^(y-1)$

The partial derivative of u with respect to y is:

$delu/dely = x^y ln(x)$

The second partial derivative of u with respect to x is:

$del^2u/delx^2 = y(y-1)x^(y-2)$

The second partial derivative of u with respect to y is:

$del^2u/dely^2 = x^y ln(x)$

The third partial derivative of u with respect to x and y is:

$del^3u/delx^2dely = y(y-1)x^(y-2) ln(x)$

The third partial derivative of u with respect to x, y, and x is:

$del^3u/del x del y del x = y(y-1)x^(y-2) ln(x)$

As you can see, $del^3u/delx^2dely = del^3u/del x del y del x$

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