Find all the second partial derivatives. f(x,y)=x4y−2x5y2 f×(x,y)= fxy(x,y)= fyx(x,y)= fyy(x,y)=

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sodni3

2021-05-09

Find all the second partial derivatives.

f (x, y) = x^{4} y - 2 x^{5} y^{2}

f_{\times} (x, y) =

f_{x y} (x, y) =

f_{y x} (x, y) =

f_{y y} (x, y) =

bahaistag

Skilled2021-05-10Added 100 answers

Step 1
Given function is

f (x, y) = x^{4} y - 2 x^{5} y^{2}

Obtain the first partial derivative with respect to x and y.

f_{x} (x, y) = {(x^{4} y - 2 x^{5} y^{2})}^{'}

= 4 x^{3} y - 10 x^{4} y^{2}

f_{y} (x, y) = {(x^{4} y - 2 x^{5} y^{2})}^{'}

= x^{4} - 4 x^{5} y

Step 2
Compute the second derivative.

f_{\times} (x, y) = {(4 x^{3} y - 10 x^{4} y^{2})}^{'}

= 12 x^{2} y - 40 x^{3} y^{2}

f_{x y} (x, y) = {(4 x^{3} y - 10 x^{4} y^{2})}^{'}

= 4 x^{3} - 20 x^{4} y

f_{y x} (x, y) = {(x^{4} - 4 x^{5} y)}^{'}

= 4 x^{3} - 20 x^{4} y

f_{y y} = {(x^{4} - 4 x^{5} y)}^{'}

= 0 - 4 x^{5}

= - 4 x^{5}

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