find all the second - order partial derivatives of the function ( f(x,y)=5x^{2}+4y + 7x^{2}y^{2} ).\n(…

find all the second - order partial derivatives of the function ( f(x,y)=5x^{2}+4y + 7x^{2}y^{2} ).\n( \frac{partial^{2}f}{partial x^{2}}=10 + 14y^{2} )\n( \frac{partial^{2}f}{partial ypartial x}=28xy )\n( \frac{partial^{2}f}{partial y^{2}}=14x^{2} )\n( \frac{partial^{2}f}{partial xpartial y}=)

find all the second - order partial derivatives of the function ( f(x,y)=5x^{2}+4y + 7x^{2}y^{2} ).\n( \frac{partial^{2}f}{partial x^{2}}=10 + 14y^{2} )\n( \frac{partial^{2}f}{partial ypartial x}=28xy )\n( \frac{partial^{2}f}{partial y^{2}}=14x^{2} )\n( \frac{partial^{2}f}{partial xpartial y}=)

Answer

Explanation:

Step1: Find the first - order partial derivative with respect to (y)

We have (f(x,y)=5x^{2}+4y + 7x^{2}y^{2}). Using the power rule (\frac{\partial}{\partial y}(ax^{n}y^{m})=amx^{n}y^{m - 1}), (\frac{\partial}{\partial y}(5x^{2}) = 0), (\frac{\partial}{\partial y}(4y)=4), (\frac{\partial}{\partial y}(7x^{2}y^{2})=14x^{2}y). So, (\frac{\partial f}{\partial y}=4 + 14x^{2}y).

Step2: Find the partial derivative of (\frac{\partial f}{\partial y}) with respect to (x)

Using the power rule (\frac{\partial}{\partial x}(ax^{n}y^{m})=anx^{n - 1}y^{m}), (\frac{\partial}{\partial x}(4)=0), (\frac{\partial}{\partial x}(14x^{2}y)=28xy).

Answer:

(28xy)