Expert Answer to "Find all first and the second partial derivatives...."

Nannie Mack

Answered question

2021-03-02

Find all first and the second partial derivatives.

f (x, y) = 2 x^{5} y^{2} + x^{2} y

Answer & Explanation

Leonard Stokes

Skilled2021-03-03Added 98 answers

Step 1 Let us find first and second order partial derivatives.
First order partial derivatives:
$f_{x} = \frac{\partial f}{\partial x} f_{y} = \frac{\partial f}{\partial y}$
Second order partial derivatives:
$f_{x x} = \frac{\partial^{2} f}{\partial x^{2}} = \frac{\partial}{\partial x} (\frac{\partial}{\partial x}), f_{y y} = \frac{\partial^{2} f}{\partial y^{2}} = \frac{\partial}{\partial y} (\frac{\partial}{\partial y})$
Step 2 Answer will be at end of step 2
$f (x, y) = 2 x^{5} y^{2} + x^{2} y$
First ordrer partial derivative:
$f_{x} = \frac{\partial f}{\partial x} = 2 . (5) x^{5 - 1} y^{2} + 2 x^{2 - 1} y$
$f_{x} = 10 x^{4} y^{2} + 2 x y$
$f_{y} = \frac{\partial f}{\partial y} = 2 x^{5} . (2) y^{2 - 1} + x^{2}$
$f_{y} = 4 x^{5} y + x^{2}$
Second order partial derivative:
$f_{x x} \frac{\partial}{\partial x} (10 x^{4} y^{2} + 2 x y)$
$f_{x x} 4 . (10) x^{3} y^{2} + 2 y$
$f_{x x} = 40 x^{3} y^{2} + 2 y$
$f_{y y} = \frac{\partial}{\partial y} (4 x^{5} y + x^{2})$
$f_{y y} = 4 x^{5}$
Step 3
Result:
$f_{x} = 10 x^{4} y^{2} + 2 x y$
$f_{y} = 4 x^{5} y + x^{2}$
$f_{x x} = 40 x^{3} y^{2} + 2 y$
$f_{y y} = 4 x^{5}$

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