Resolved: "Given f(x,y)=2x2−xy3+4y6, find fxx(x,y)= fxy(x,y)="

preprekomW

Answered question

2021-06-07

Given $f (x, y) = 2 x^{2} - x y^{3} + 4 y^{6}$ , find
$f_{x x} (x, y) =$
$f_{x y} (x, y) =$

Answer & Explanation

falhiblesw

Skilled2021-06-08Added 97 answers

Step 1
Consider the given function.
$f (x, y) = 2 x^{2} - x y^{3} + 4 y^{6}$
The objective of the question is to find the value of $f_{x x} (x, y) and f_{x y} (x, y)$ .
These are the second order partial derivatives.
Step 2
Partially differentiate the function with respect to x.
$f_{x} (x, y) = 2 \cdot 3 x^{3 - 1} - 1 \cdot y^{3} + 0$
$= 6 x^{2} - y^{3}$
Partially differentiate again with respect to x to find $f_{x x} (x, y)$ .
$f_{x x} (x, y) = 6 \cdot 2 x - 0 = 12 x$
Partially differentiate the function with respect to y.
$f_{y} (x, y) = 0 - x \cdot 3 y^{3 - 1} + 4 \cdot 6 y^{6 - 1}$
$= - 3 x y^{2} + 24 y^{5}$
Partially differentiate again with respect to x to find $f_{x y} (x, y)$ .
$f_{x y} (x, y) = - 3 y^{2} \cdot 1 + 0$
$= - 3 y^{2}$
Thus, the required partial derivatives are obtained.
$f_{x x} (x, y) = 12 x$
$f_{x y} (x, y) = - 3 y^{2}$

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