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}}=square )

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}}=square )

Answer

Explanation:

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

We use the power rule (\frac{\partial}{\partial x}(x^n)=nx^{n - 1}) and treat (y) as a constant. For (f(x,y)=5x^{2}+4y + 7x^{2}y^{2}), (\frac{\partial f}{\partial x}=\frac{\partial}{\partial x}(5x^{2})+\frac{\partial}{\partial x}(4y)+\frac{\partial}{\partial x}(7x^{2}y^{2})) (\frac{\partial f}{\partial x}=5\times2x+0 + 7\times2xy^{2}=10x + 14xy^{2})

Step2: Find the second - order partial derivative with respect to (x)

Differentiate (\frac{\partial f}{\partial x}=10x + 14xy^{2}) with respect to (x) again. (\frac{\partial^{2}f}{\partial x^{2}}=\frac{\partial}{\partial x}(10x)+\frac{\partial}{\partial x}(14xy^{2})) Using the power rule (\frac{\partial}{\partial x}(ax^{n})=anx^{n - 1}) ((a = 10,n = 1) and (a = 14y^{2},n = 1)), we get (\frac{\partial^{2}f}{\partial x^{2}}=10+14y^{2})

Answer:

(10 + 14y^{2})