Best solutions to - "Find all second-order partial derivatives for f(x,y)=−4x3−3x2y3+2y2."

osteoblogda

Answered question

2022-01-17

Find all second-order partial derivatives for

f (x, y) = - 4 x^{3} - 3 x^{2} y^{3} + 2 y^{2}

Maria Lopez

Beginner2022-01-18Added 32 answers

Step 1
To find all second-order partial derivatives for

f (x, y) = - 4 x^{3} - 3 x^{2} y^{3} + 2 y^{2}

.
Step 2
Let us compute the first-order partial derivatives.

f_{x} (x, y) = - 4 (3 x^{2}) - 3 y^{3} (2 x) + 0 = - 12 x^{2} - 6 x y^{3}

f_{y} (x, y) = 0 - 3 x^{2} (3 y^{3}) + 2 (2 y) = - 9 x^{2} y^{3} + 4 y

Step 3
So the second-order partial derivatives,

f_{\times} (x, y) = - 12 (2 x) - 6 y^{3} (1) = - 24 x - 6 y^{3}

f_{y x} (x, y) = 0 - 6 x (3 y^{2}) = - 18 x y^{2}

f_{x y} (x, y) = - 9 y^{3} (2 x) + 0 = - 18 x y^{2}

f_{y y} (x, y) = - 9 x^{2} (3 y^{2}) + 4 (1) = - 27 x^{2} y^{2} + 4

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