Solved: "Calculate the second-order partial derivatives off (x,y)=x3+y2ex"

College
Calculus and Analysis
Differential Calculus
Derivatives

Shelia Lawrence

Answered question

2021-12-10

Calculate the second-order partial derivatives off

(x, y) = x^{3} + y^{2} e^{x}

Answer & Explanation

temnimam2

Beginner2021-12-11Added 36 answers

Step 1
To calculate the second order partial derivatives of the function $f (x, y) = x^{3} + y^{2} e^{x}$
Step 2
Let us calculate the first order partial derivatives.
$f_{x} (x, y) = 3 x^{2} + y^{2} e^{x}$
$f_{y} (x, y) = 0 + e^{x} (2 y) = 2 e^{x} y$
Step 3
Now we calculate the second order partial derivatives.
$f_{x x} (x, y) = 3 (2 x) + y^{2} e^{x} = 6 x + y^{2} e^{x}$
$f_{x y} (x, y) = \frac{\partial (f_{y} (x, y))}{\partial x} = 2 e^{x} y$
$f_{y x} (x, y) = \frac{\partial (f_{x} (x, y))}{\partial y} = 0 + (2 y) e^{x} = 2 e^{x} y$
$f_{y y} (x, y) = \frac{\partial (f_{y} (x, y))}{\partial y} = 2 e^{x}$

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