Find $f_{x x}, f_{x y}, f_{y x}$ and $f_{y y}$ for the following function. (Remember, $f_{y x}$ means to differentiate with respect to $y$ and then with respect to $x$ .)
$f (x, y) = 9 y \ln x$
$f_{xox} = ◻$

Answer

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Answer

$f_{x x} = - \frac{9 y}{x^{2}}, f_{x y} = \frac{9}{x}, f_{y x} = \frac{9}{x}, f_{y y} = 0$

Steps

Step 1 ：First, we find the first order partial derivatives of the function $f (x, y) = 9 y \ln x$ .

Step 2 ：We find that $f_{x} = \frac{9 y}{x}$ and $f_{y} = 9 \ln x$ .

Step 3 ：Next, we differentiate $f_{x}$ with respect to $x$ to get $f_{x x}$ , and we find that $f_{x x} = - \frac{9 y}{x^{2}}$ .

Step 4 ：We then differentiate $f_{x}$ with respect to $y$ to get $f_{x y}$ , and we find that $f_{x y} = \frac{9}{x}$ .

Step 5 ：We differentiate $f_{y}$ with respect to $x$ to get $f_{y x}$ , and we find that $f_{y x} = \frac{9}{x}$ .

Step 6 ：Finally, we differentiate $f_{y}$ with respect to $y$ to get $f_{y y}$ , and we find that $f_{y y} = 0$ .

Step 7 ： $f_{x x} = - \frac{9 y}{x^{2}}, f_{x y} = \frac{9}{x}, f_{y x} = \frac{9}{x}, f_{y y} = 0$