Question

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

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:

\(28xy\)