We have an answer to "Write first and second partial derivatives f(x,y)=2xy+9x2y3+7e2y+16 a)fx..."

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Joni Kenny

Answered question

2021-02-02

Write first and second partial derivatives
$f (x, y) = 2 x y + 9 x^{2} y^{3} + 7 e^{2 y} + 16$
a) $f_{x}$
b) $f_{x x}$
c) $f_{x y}$
d) $f_{y}$
e) $f_{y y}$
f) $f_{y x}$

Answer & Explanation

unett

Skilled2021-02-03Added 119 answers

a) $\frac{\partial f}{\partial x} = \frac{\partial}{\partial x} [2 x y + 9 x^{2} y^{3} + 7 e^{2 y} + 16]$
$f_{x} = 2 y + 18 x y^{3} + 0 + 0 = 2 y + 18 x y^{3}$
$\Rightarrow f_{x} = 2 y + 18 x y^{3}$
b) $f_{x x} = \frac{\partial f_{x}}{\partial x} = \frac{\partial}{\partial x} [9 y + 18 x y^{3}]$
$= 0 + 18 y^{3} = 18 y^{3}$
$f_{x x} = 18 y^{3}$
с) $f_{x y} = \frac{\partial f_{x}}{\partial y} = \frac{\partial}{\partial y} [2 y + 18 x y^{3}]$
$f_{x y} = 2 + 18 x \cdot 3 y^{2}$
$f_{x y} = 2 + 54 x 6^{2}$
d) $f_{y} = \frac{\partial f}{\partial y} = \frac{\partial}{\partial y} [2 x y + 9 x^{2} y^{3} + 4 e^{2 y} + 16]$
$f_{y} = 2 x + 9 x^{2} \cdot 3 y^{2} + 7 \cdot 2 e^{2 y} + 0$
$f_{y} = 9 x + 27 x^{2} y + 14 e^{2} y$
e) $f_{y y} = \frac{\partial}{\partial y} [2 x + 27 x^{2} y^{2} + 14 e^{2 y}$
$= 0 + 27 x^{2} x^{y^{2}} + 14 \cdot 2 e^{2 y}$
$f_{y y} = 54 x^{2} \cdot 2 y + 28 e^{2 y}$
f) $f_{y x} = \frac{\partial f_{y}}{\partial x} = \frac{\partial}{\partial x} [2 x + 27 x^{y^{2}} + 14 e^{2 y}]$
$f_{y y} = 2 = 54 x y^{2}$

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